QA 310 






.VJ^\F* .V ^^ 












A TABLE 

OF 

INTEGRALS 



BY 

RALPH G. HUDSON 

AND 

JOSEPH LIPKA 



The following tables are taken from the Mathematical Section 
of the Engineers* Manual in which are consolidated the principal 
formulas and tables of Mathematics, Mechanics, Hydraulics, 
Heat and Electricity. The complete Mathematical Section is 
published under the title, A Manual of Mathematics. 



CopYiaGHT, 191 7, 

BY 

RALPH G. HUDSON 

AND 

JOSEPH LIPKA 



APR 18 1917 



Stanbopc pv<iS2 

H. GILSON COMPANT 
BOSTON, U.S.A. 



A TABLE 

OF 

INTEGRALS 



BY 

RALPH G. HUDSON, S.B., 

Assistant Professor of Electrical Engineering at the 

Massachusetts Institute of Technology 

AND Harvard University 

AND 

JOSEPH LIPKA, Ph.D., 

Instructor in Mathematics at the Massachusetts 
Institute of Technology 



FIRST EDITION 
first thousand 



NEW YORK 

JOHN WILEY & SONS, Inc. 

London: CHAPMAN & HALL, Limited 

1917 



v/c— 



Table of Derivatives 



TABLE OF DERIVATIVES 

I 

i 



n* 



Functions of x are represented by u and v, constants are represented by a, 
ii, and e. 



d / . . N du , dv , 

^(u±v±...)=^±^±---. 

d , , du 
■J- (au) = a.^-. 
dx ^ ^ dx 

d , . dv , du 

-r- (UV) = U 3- + V 3-- 

dx^ ^ dx dx f 

du dv d . du 

J / X V^i U-r- -r-SinU = COSU-J-. 

d /u\ dx dx dx dx 



(;) 



dxVvy v2 d . du 

A Au :5-cosu = — sinu^-. 

a /-nN n-1 ^^ dx dx 



dx ^ ^ dx 



d , , du 

1 J 3-tanu = sec^u^i-. 

d , loga e du dx dx 



dx udx 



d du 



d , du 3- sec u = sec u tan u -3- . 

^a" = a"lna^. dx dx 

dx dx 

J J d du 

d ,, ydu -T-cscu = — cscucotu 



dx dx 

iLu^ = vu--^ + uMnu^. ^versu = sinu^. 



dx" " dx 

d du du d . du 

" ,,v _ „„v— 1 ^*~ _i_ «v In 11 !zi: — vers u = sin u -^- 



— sin-i u = ^ ^ 1 where sin-^ u lies between - - and + - ) ° 
dx Vi - u2 dx \ 2 2 / 

— cos-i'ii'^* ^ -^ (where cos"^ u lies between o and it). 

dx -.. Vi — u^dx 

! d ^ , I du 

, 3-tan-Vu = — i — 5 J-. 
I dx I + u2 dx 

: d ^, I du 

i dx I + u^ dx 

' — sec-i u = ^ -3^ (where sec-^ u lies between and it). 

j dx u Vu2 — I dx 

^ — csc-^ u = ' -^ ( where csc-^ u lies between - - and + - )• 

dx u Vu2 - I dx \ 2 ^ / 

— vers-i u = ^ -^ (where vers"^ u lies between p and it). ^^ 

;'^"-"^'^ ©GIA457994 <^ ^ iC 



Functions containing ax + b 3 

TABLE OF INTEGRALS 
Fundamental Forms 

Jdf(x) =f(x)+C. 

d rf(x)dx = f(x)dx. 
J[fi(x) ± f2(x) ± • . . ] dx = Jfi(x) dx db Jf2(x) dx ± . . . . 

j a f (x) dx = a j f (x) dx, where a is any constant. 
u^ du = — — h C (n 7^ — i) ; u is any function of x. 

j — • = lnu + C; uis any function of x. 

j u dv = uv — j V du; u and v are any functions of x. 

Note. In the following ^able, the constant of integration (C) is omitted 
but should be added to the rd;sult of every integration. The letter x represents 
any variable; the letter u represents any function of x; all other letters repre- 
sent constants which may have any finite value unless otherwise indicated; 
In = lege; all angles are in radians. 

Functions containing ax + b 

1 /(ax + b)- dx = ^7(^qr7y (ax + h)^K (n ?^ -i) 

2 /axT-b = ^^^^^ + ^)- 

3 /x(ax + b)" dx = ^^^ (ax + b)^^ - ^.^ (ax + b)«+^ 

(n?^-i, -2) 
. r xdx X h . , , ,. 

5 fl^^fw -= a^axV b) + i^ '" <" + ^^- 

J ^ a^Ln + s n + 2 n + ij 

(n ^ -I, -2, -3) 

■^ /^f^ == ^3 Q (ax + b)2 - 2 b(ax + b) + b^ In (ax + b)] • 

s /sllV^=^[^^^+^^-^^^^^^^+^^-^b]- 

Q f x'dx I r, , , ., , 2b b^ 1 

9 J(5^+bF-^r<^ + ''> + ^r+b- 2(ax + b)d - 



Functions containing ax + b 
10 J'x"»(ax + b)"dx 

= . a(m+'n + i) [^"^(ax+b)-Hi-mb/x-i(ax+b)"dx] 

J x(ax + b) b ax + b* 
9 f dx ^la ax + b 
J x2(ax + b) bx"^b2'^ x 

dx I I , ax + b 



In 



J x(ax + b)2 ~ b(ax + b) b^ *" x 
4. r dx b + 2ax ■ 2a^ ax + b 

* J x2(ax + b)2 ~ b2x(ax + b) "^ b^ *^ x 

K r ^^ I , Vax + b - Vb ,, ■ 

^ J xV^7+b = VE ^^ V^^+b + Vb' ^^ P"'-^ 

/. /• <ix 2 /ax + b /, X , 

6 I / — - = -j== tan-i 1/ — ^- (b neg.) ; 

^xvax + b V— b V "b 

7 r ^^ -=. ? —-^-( ^^ ^ . /noddand\ 

-^ x(ax + b)2 b(n - 2) (ax + b)^"' ^^ x(ax + b)^"' \ Pos- / 

Q f Vax + b , / -^ , /r, Vax+b — Vb ,, . 

8 j— ^dx = Wax + b + Vbln-^==^-^. (bpos.) 

9 /^^^±5dx = 2VS?+b-2V3btan-\/?^- (bneg.) 

20 / i^±vl dx = 2 (ax + b)i + b / <^ +">'"' dx. (n odd and pos.) 

rt- f dx Vax + b a , Vax + b - Vb ., . 

21 I — , = ^ 7= In , ^' (b pos.) 

J x2 Vax + b bx 2 b Vb Vax + b + Vb 

22 r_^_.=-^^^±5-^=tan-V^^- (bneg.) 
»^x2Vax + b bx bV-b ^ -b 

23 /(ax + b?(px + q)=bT^'°gT^- ("P-^''^") 

2* J(SqTOfi)=bF^[sTb+bFi^'»sfb]- ("P-^"'^") 

^^ J (ax + b)^(px + q)^ " (m - i) (bp - aq) L(ax + b)^-Mpx + q)"^-^ 

- a (m + n - 2) /^^ _^ ^^^, (px + q)-J * 
(m > I, n pos., bp — aq 5^ o) 

26 / (ax+b)?px+q) = b-^G^-- ^^ + ^^ -p-^- ^'^^^^' (^^-^^^0) 

27 r ^^^ = ' r - i_-inP^+-9l. 

^' J (ax+b)2 (px+q) bp - aqL a(ax + b) bp - aq ax + bj 

(bp — aq 7«^ o) 



Functions containing ax^ + b 6 

28 /v^^+1^^^ = ^ (3 aq - 2 bp + apx) Vax + b. 

29 r^^^+^dx = l^^^±^ - gi/aq-bp^^^_,i/p(ax + b)^ 
Jpx + q p P'^P taq — bp 

(P p os., aq > b p) 

30 r ^ax + b ^^ ^ 2 Vax + b ^ i t/ bp - aq ^^ Vp(ax+b)- Vbp-aq 
»^PX + q P pVp Vp(ax+b) + Vbp-aq* 

(P pos., bp > aq) 

/- dx ^ I ^ ^ Vp(ax + b) - Vbp - aq 

J (px + q)Vax + b Vp Vbp - aq Vp(ax + b) + Vbp - aq 

(p pos., bp > aq) 

33 r^^dx = 1 V (a. + b) (px + q) - ^^^ 
^ Vax + b a ,^ av^ap 

In (Vp(ax + b) + Va(px + q)). (a and p, same sign) 

1^/7 j— irTT \ — ^ bp — aq, _ V-ap(ax+b) 

= - V(ax + b) (px + q) - . ^ tan i / 

a av— ap avpx-fq 

(a and p have opposite signs) 

= iV(ax + b)(px + q)+-^P^ 
a 2 av — ap 

. _, 2 apx + aq + bp , , , . . v 

sin 1 r- — — -• (a and p have opposite signs) 

Functions containing ax^ + b 

oc r dx I , xVa — V— b , , . 

35 I — ^ I , = — / In — — -. (a pos., b neg.) 

Jax2+b 2V-ab xVa + V-b ' ^ ' ^^ 

, Vb+xV^ ^ 

In — ;= (a neg., b pos.) 



2 V— ab Vb — xV— { 

QR f dx ^ I X 2n-3 /» dx /ninteg.\ 

J(ax2 + b)" 2(n-i)b(ax2+b)"-i"^2(n-i)bJ (ax2+b)"-i V >i / 

37 /(ax^ + b)"xdx = i^^5^±^\ (n^-x) 

38 /^ = fa^-^-^+^)• 
■^^ J x(ax2 + b) 2 b '" ax2 + b 
4.0 f x'^ dx _ X b /" dx 
*" Jax2 + b~a aJax2 + b* 



5 Functions containing ax^ + b 

4.1 f x2 dx ^ I X . I r dx 

J (ax2 + b)^ 2(n - i)a (ax^ + b)"-i "*" 2(n - i)a J (ax^ + b)"-i" 

(n integ. > i) 
An r dx If dx a r dx , . ^ . 

^2 J xHax^ + b)- = b J xHax2 + b)-i ~ b J (ax^+b)-' ^"^ P^'' ^"^^^'^ 

43 r ^ax2 + b dx = - Vax2 + b + — 7= In (xVa + Vax2+b) . (a pos.) 
J 2 2 Va 

44 JV^^M=1bdx = ^V^^r+b + --^sin-i^xy/_^]. (aneg.) 

45 f , ^^ = -4^ In (x Vi + Vax2 + b) . (a pos.) 
J V ax2 + b V a 

47 fVax^ + b X dx = — (ax2 + b)^- 

J SSL 

Jx dx T ^ 

-7==== = iVax2+b. 
V ax2 + b a 

49 /:^^?|±1 dx = VS?+b + Vb to ^^-^g . (b pos.) 

50 r^^??+5dx = V^^+b-Vrbtan-^^^±I. (bneg.) 
J X V— b j 

^ dx I , Vax2 + b - v^ ,u N 

Ki ( — , = —^ In (b pos.) 

^^ J xVax^ + b Vb X 

52 f — / ^ — = / sec-Mxi/-M- (bneg.) 

^^ JxVax2 + b V-b V V V 

63 rVax2+b x2dx = ^(ax2 + b)^ -^Vax^ + b 
J 4 a o a 

7^ In (x Va + Vax2 + b) . (a pos.) 

8ava 

54 fV^~+h x2 dx = — (ax2 + b)^ - ^ Vax2 + b 
^ 4 a o a 

"8aV3^^^^"(V^')- ^^"^^-^ 

55 f /'^^ = — Vax2 + b ^In (x Va + Vax2 + b) • (a pos.) 

J Vax2 + b 2 a 2 a Va 

56 f Z''^'' = — Vax2+b -V=sin-i(x\/-^V (aneg.) 

^'^JVax2 + b 2 a 2aV-a \ V b/ 

5^ jV^|±b^^ ^ _V^|+b ^ Valn(xVa + V^^+h) . (a pos.) 

-o fVIx2 + b , Vax2 + b ^/ — . ,( i/~a\ , . 

68 J ^r^<^^ = ^ ^~^ s^^ V^ V ~b/ • (^ ^^s) 

/^ dx ^ Vax2 4- b 

^^ Jx2V^^Mn^ bx 



Functions containing ax^ + bx + c 



^ x"dx _ x"-^ V ax^ 4- b _ (n - i) b r x^-^dx 
^^ J V^^^+b " na na J VW+h' ^^ P^''^ 

61 pVi^dx^^-!^!!^ 

(n pos.) 

«„ r Vax^ + bdx (ax^ + b )i (n - 4) a V ^^ax^+b 

62 J ^ - b( n-i)x'^ (n-i)bJ x-2 ^- ^^ P^'-^ 

^ dx _ Vax^ + b _ (n— 2) a r dx 

6^ J x-V^^Tb ~ ~b (n - I) x-i (11^:^)^ J x-2 V^^cH^' ^^P^"') 

64 J(ax2 + b)^ dx = I (2 ax2 + 5 b) Vax^ + b 

3b2 _ 

+ ^-;y^ In (x Va + Vax^ + b) . (a pos.) 

65 J(ax2 + b)^dx = I (2 ax2 + 5 b) Vax^ + b 

66 r^^?_^ = _^=. 

^ (ax2 + b)2 b Vax2 + b 

67 r (ax2 + b)^ X dx = — (ax2 + b)l 
J 5 a 

xdx I 



68 f^^^= y^=- 

^ (ax2 + b)^ a Vax2 + b 



69 f— ^^^ = T^ + -Vln (x Va + Vax2 + b). (a pos.) 

^ (ax2 + b)^ a V ax2 + b a v a 

70 r^!d^^_ ^ + ^_3in-^Li/-^y (aneg.) 

-'(ax2+b)^ aVax2 + b a V-a \ V b/ ' 

dx I , x" 

In 



71 f ^? = 2_ 

' J X (ax'^ + b) bn '" ax" + b 

dx _ 

X Vax" +b ~ n Vb "* Vax'^ + b + Vb 



_„ f dx I , Vax" + b-Vb ,, 

72 — ., = — 7= In ■ j=- (b pos.) 



73 f ^j! =_^sec-^V^. (bneg.) 

*^xVax"-|-b nV-b ^ b 

Functions containing ax^ + bx + c 



„A r dx I , 2 ax + b — Vb2 — 4 ac ., „ ^ . 

74 I „ , , — , — = r In ^ ^' (b2 > 4 ac) 

J ax2 + bx + c Vb2 - 4 ac 2 ax + b + ^b^ - 4 ac 

75 r ,^t -u = .— tan- ;^ + ^ . (b2 < 4 ac) 

J ax2 + bx + c V4 ac - b^ V4 ac - b^ 



8 Functions containing sin ax 

„o r x2 dx X b -,,,,,,, b2 — 2 ac /- dx 



ax2 + bx + c 



79 ( , ^ = = -4=ln(2 ax + b + 2 Va Vax^ + bx + c). (a pes.) 
-^ Vax2 + bx + c V a 

on C dx I . _, — 2 ax — b , , 

80 1 - ■= = —^= sin-i - , (a neg.) 
♦^ V ax2 + bx + c V -a v b2 - 4 ac 

81 fV ax^+bx+c dx^g^^^ V ax2+bx+c + i^|^ r^^^-= 

J 4a 8a J Vax2 + bx+c 

r%n C X dx _ Vax^ + bx + c _ _b^ r dx 

-^ Vax2 + bx + c ~ a 2a J Vax^ +bx + c* 

83 rVax2 + bx + c X dx = (^^' + ^x+c)- _ ^ r Vax2 + bx + c dx. 
J 3 a 2 a J 

Q/i r dx I / Vax2 + bx + c + Vc , b \ , 

84: I — , = = 7^ In 1 7=)- (epos. 

J X Vax2 + bx + c Vc V ^ 2 Vc/ 

oc r dx I . . bx + 2 c . . 

85 I — , — = -7= sin-i . (c neg.) 

•^xvax2 + bx + c v-c xvb^ — 4ac 

86 r z^"" =-^Vax2 + bx. 
-^ X Vax2 + bx DX 

87 r dx ^ 2 (2 ax + b) 

•^ (ax2 + bx + c)^ (b2 - 4 ac) Vax2 + bx + c 



Functions containing sin" ax 

88 J sin u du = —cos u. (u is any function of x) 

89 1 sin ax dx = — cos ax. 
J a 

r\r\ r ' A X sin 2 ax 

90 \ sin2 ax dx = - 



4a 



91 j sin^ ax dx = — cos ax H cos^ ax. 

J a 3a 

92 fsin^ ax dx = |. X sin 2 ax H sin 4 ax. 

J 8 4a '32a^ 

e\n C ' r> J sin^~^ ax cos ax . n — i r . „ „ , , . ^ . 

93 I sin" ax dx = 1 | sm"-2 ax dx. (n.pos. integ.) 

J na n ./ 

^- /* dx I , ^ ax I , , . s 

94 I ^ = - In tan — = - In (esc ax — cot ax). 

^ sm ax a 2 a 

95 f . f = — cot ax. 

J sin2ax a 

ft/? f dx I cos ax , n — 2 /• dx / • ^ ^ ,\ 

97 f dx it^(-_axy 

J I + sin ax a \4 2 / 



Functions containing cos ax or sin ax and cos ax 9 

98 f^*^ = icot(^-!?). 

Ji— smax a \4 2/ 

99 r d^. ,_^^tau-r\/^'tan(^-g)1. (b» > c^) 
Jb + csinax a Vb^ — c^ |_Tb + c V4 2 /J 

100 r ^^ ^ -I ^^ c+bsinax+Vc2-b^cosax ^^2 > 1,2) 
J b + c sin ax ^ Vc^ — b^ b + c sin ax 

1 ni C ' • V J sin (a - b) X sin (a + b) x / , , i^ox 

101 I sin ax sm bx dx = — ) rr 7 — r-uf- • (a^ 7^ b^) 

J 2 (a — b) 2 (a + b) 

Functions containing cos ax 

102 j cos u du = sin u. (u is any function of x) 

103 j cos ax dx = - sin ax. 

1 Ail C 9 J X , sin 2 ax 

104 I cos2 ax dx = - H 

J 24a 

105 I cos^ ax dx = - sin ax sin^ ax. 

J a 3a 

106 I cos^ ax dx = J X -^ sin 2 ax H sin 4 ax. 

J 84a 32 a ^ 

1 AT r r, J cos"-i ax sin ax, n — If „_, - , . ^ . 

107 I cos'^ ax dx = ■ H I cos"^ ax dx. (n pos. integ.) 

108 1 =-lntan( h-|= -In (tan ax + sec ax). 

Jcosaxa \24/a 

109 f 



dx I 



tan ax. 



cos^ ax a 
dx I sin ax , n — 2 r dx 



AACi C ^^ I sin ax , n — 2 f ax , . ^ ^ . 

"2 Jr 



dx I ^ ax 

= - tan — 

cos ax a 2 

dx I ^ax 

= — cot 



cos ax a 2 



113 f— i5 = ,1 tan-(V^tang). (b^ > c^) 

J b + c COS ax a Vb^ — c^ \Vb + c 2 J 

114 fi— ^^— = ' In " + " "" f^ + ^^'^^' °' -£g. (c» >b») 

J b + c COS ax a Vc^ — b^ b + c cos ax 

1 1 K r V J sin (a — b) X , sin (a + b) X , , _^ vox 

115 I cos ax cos bx dx = — ) t^. — 7 — , ; . (a^ 9^ b^) 

J 2 (a — b) 2 (a + b) ' 

Functions containing sin ax and cos ax 

1 1 c r • u J I fcos (a — b) X , cos (a + b) x"! , , . .ox 

116 J sin ax cos bx dx = — —h — ' ', . {a? 9^ b^) 

117 I sin'^ ax cos ax dx = , ^ . — ; sin"+i ax. (n 7^ — i). 

J a(n+i) 

mf cos ax , I , . 
I dx = - In sin ax. 

J sinax a 



10 Functions containing sin ax and cos ax 

119 r (b + c sin ax)^ cos ax dx = — , ^ (b + c sin ax)'*+^ (n ?^ — i) 
J ac (_n ~\- 1 ) 

iOA f cosaxdx I 1 /u I • N 

120 I 7—, r = — In (b + c sm ax). 

J b + c sin ax ac 

121 I cos" ax sin ax dx = -. — ; — r cos'^+i ax. (n 5^ — i ). 

J a (n + I ) ^ ' 

1 00 fsin ax , i , 

lz2 I dx = — In cos ax. 

J cos ax a 

123 r(b + c cos ax)'^ sin ax dx = — ^ — - (b + c cos ax)'H-i. (n ^^^ _ i ) 

J ac (n + i; 

124 r sin ax — ^^ ^ _\_ jj^ ^^ _|_ ^ ^^g ^^^ 
^ b + c cos ax ac 

125 ^r-. ^ = / Inftan^fax + tan-^^)]^ 

^ b sin ax + c cos ax ^ Vb^ + c2L 2V b/J 

ioff r • 9 9 J X sin 4 ax 

126 I sin^ ax cos^ ax dx = - — - — 

J 8 32 a 

127 f- — = - In tan ax. 

J sin ax cos ax a 

129 r?iS!Hdx = ir-sin ax + In tan(?5 + ![)]. 
J COS ax a L \ 2 4 / J 

^ oA rcos2 ax If ax"l 

130 I -; dx = - cos ax + In tan — • 

J sin ax aL 2 J 

^oi r • ™ ., J sin"'-! ax cos'^+i ax 

131 I sin*" ax cos" ax dx = -. ; — r 

J ^ a (m + n) 



132 / 



sin*" ax cos" ax dx = 



H ; 1 sin'"-^ ax cos" ax dx. (m, n pos.) 

m -j- n ./ 

sjnw+i ax cos"-! ax 



a (m + n) 

H ; — jsin'" ax cos"-^ ax dx. (m, n pes.) 

m + n 1/ 

loq /" cos" ax . _ — cos"+!ax 

^^^ J sin"' ax a (m - I) sin"'-i ax 

, m — n — 2 r cos" ax , . ^ . 

+ —, ^ I . m 9 — dx. (m, n pos., m 5^ i) 

(m — i) J sin"'-2ax 

134 / 



sin"' ax , sin"'+! ax 



dx = 



cos" ax a (n — I ) cos"-! ax 

m— n+2 c sin"'ax , , , . 

I — -zTh — dx. (m, n pos., n 7^ i) 

n - I J cos"-2 ax v » h » 

135 pin^^ax ^^ ^ Wi -cos^ax)" ^^^ (Expand, divide, and use 103-108) 
J cos ax J cos ax 

136 rcos^:^ ^^ ^ r (i - sin^ax)" ^^ (Expand, divide, and use 89-94) 
J smax J sin ax 



Functions containing tan ax, cot ax, sec ax, or esc ax 11 

^o- /»sin2^iax, /-(i- cos2ax)^ . , 

137 I dx = I ^ ^smaxdx. 

J cos ax J cos ax 

(Expand, divide, and use 121-122) 

^oo rcos2"+iax, /•(! - sin2ax)^ . 

138 I — -. dx = I ^^ -. — cos ax dx. 

J sin ax J smax 

(Expand, divide, and use 117-118) 

Functions containing tan ax ( = — - — ; or cot ax ( =7 ) 

V cot ax/ V tan ax/ 

139 ) tan u du = —In cos u. (u is any function of x) 

140 ftan ax dx = — In cos ax. 

141 ftan^ ax dx = - tan ax — x. 

142 ftan^ ax dx = , ^_ . tan^i ax - rtan^2 ax dx. (n integ. > i) 

143 jcot u du = In sin u. (u is any function of x) 

144 fcot ax dx = \- — ^ = - In sin ax. . 
J J tan ax a 



tan^ax 
dx 



— cot ax — X. 



145 rcot2 ax dx = f 

146 /cofaxdx = /jj^= -^^^ cot-'ax - /cot-^azdx. 

(n integ. > i) 

1^^ / b + c^cotax ==/ b'tTn^+c = b^[bx-^ln (ccosax+bsinax)]. 

149 r -7=^= = \ sin ax. 

-^ V I 4- tan2 ax a ^__ 

150 f , ^ =. = — ] sin-i ( V -^^ sin ax ). (b pos., h^ > c^) 
^ Vb+ctan^ax a v b - c \^ o J 

Functions containing sec ax ( = ) or esc ax ( = ) 

V cos ax/ V smax/ 

151 j sec udu = in (sec u+tanu)=lntan (- + -)• (u is any function of x) 

152 rsecaxdx= - Intan (— +-]• 

153 fsec^ ax dx = - tan ax. 

/I sm d.^ ti —' 2 /* 
sec^ ax dx = —, r — — 1 | sec^-2 ax dx. (n Integ. > i) 
a(n — i) cos"* lax n — i J \ & / 

155 Jcsc u du = In (esc u — cot u) = In tan - • (u is any function of x) 



2 Functions containing inverse trig, or alg. and trig, functions 

56 (esc ax dx = - In tan — • 
J a 2 

67 jcsc2axdx= — cot ax. 

,58 fcsc^ ax dx = j^ — r ^^l^ + ?^^ f csc'^-^ ax dx. (n integ. > i) 

J a(n-i) sin"-iax n— iJ ^ s^ / 

Functions containing tan ax and sec ax or cot ax and esc ax 

59 r tan u sec u du = sec u. (u is any function of x) 

60 j tan ax sec ax dx = - sec ax. 

61 ftan'' ax sec^ ax dx = , V — r- tan'H-i ax. (n 5^ — i) 
J a (n + i) 

«o rsec^axdx i , . 

62 I — : = - In tan ax. 

J tan ax a 

63 j cot u CSC u du = —CSC u. (u is any function of x) 

64 (cot ax CSC ax dx = - — esc ax. 
J a. • 

65 fcot^ ax csc2 ax dx = ^, — r cot^^ ax. (n 5^ - 1) 

J a (n + i; 

«^ fcsc^axdx I , 

66 ( 1 = — - In cot ax. 

J cot ax a 

Inverse Trigonometric Functions 

67 fsin-i ax dx = X sin-^ ax + - Vi - a?x^. 

68 fcos"! ax dx = xcos~i ax Vi — a^x^. 

69 ftan-i ax dx = X tan-i ax- — In (i + a^x^). 
J 2 a 

70 fcot-i ax dx = X cot-i ax + — In (i + a^x^). 

71 jsec"^ ax dx = X sec~i ax In (ax + Va?x^ — i). 

72 jcsc"^ ax dx = X csc~i ax + - In (ax + v^a^x^ — i). 

Algebraic and Trigonometric Functions 

73 f X sin ax dx = — sin ax x cos ax. 

J a^ a • 

74 fx" sin ax dx = — - x" cos ax + - fx'^-i cos ax dx. (n pos.) 

(ax)3 (ax)^ 
3l3 5L5 



__ r sinaxdx _ _ (ax)^ (ax)^ 
J X 3 l3 



Exponential, Algebraic, Trigonometric, Logarithmic Functions 13 

/I I 

X cos ax dx = — cos ax + - x sin ax. 
a^ a 

177 fx"^ cos ax dx = - x" sin ax j x"~i sin ax dx. (n pes.) 

178 f-^^^^ = lnax-^ + ^'-.. 

Exponential, Algebraic, Trigonometric, Logarithmic Functions 

/b" 
b" du = — r • (u is any function of x) 

180 I e" du = e". (u is any function of x) 

181 / 



b«^dx 



alnb 

182 fe"^ dx = - e°^. 
dx I 



184 /bW« "^-^ (»> + -"''• 

185 / be°4V-- = r^*^''"^V/')- (band epos.) 



xb«^ dx = 



xb 



186 j^^ ^^ ^jj^^ aMlnb) 

xe«^dx =^(ax- I). 

xn^ax^jx = ±-^ i^- x"-ib«^dx. (n pos.) 

alnb alnb J ^ t- / 

189 fx'^e^^ dx = i x"e«^ - - fx^-ie"^ dx. (n pos.) 

190 r|!dx = lnx + ax+W + M!+.... 

J X 2 [2 3 1^ 

192 fe""^ In X dx = - e«^ In x - - f— dx. 
J a aJ X 

/gca; 
e"^ sin bx dx = , , , , (a sin bx — b cos bx). 
a^ + b^ 

/goa; 
e"^ cos bx dx = ^ , , . (a cos bx + b sin bx). 
a^ + b'' 

xe°^ sin bx dx = ^ (a sin bx — b cos bx) 

- (a2 + b^)2 f (^' ~ ^') ^^° bx - 2 ab cos bx]. 



14 Some Definite Integrals 



/xe*^^ 
xe°'^ cos bx dx = — (a cos bx + b sin bx) 



+ 

qOX 

(a2 + b^)2 1(^^ "" ^^) ^°s bx + 2 ab sin bx]. 



197 I In ax dx = X In ax — x. 

198 f (In ax)« dx = X (In ax)'^ - n f (In ax)'^-i dx. (n pes.) 

199 Jx«lnaxdx = x-+.[i^-^^J. („ pos.) 

200 ffl^^dx = 2i^. („^_,) 
J X n + I 

201 f~^ = In (In ax). 
J x In ax 

o^o r dx I fi /I Nit , (lnax)2 , (Inax)^ "1 

202 , = - In (In ax) + In ax + ^ — r-^ + ^ — ^ + • • • . 

JlnaxaL 2[2'3|_3 J 

/X ' 

sin (In ax) dx = - [sin (In ax) — cos (In ax)]. 

204 r cos (In ax) dx = - [sin (In ax) + cos (in ax)]. 

Some Definite Integrals 

205 rVa^^:^dx = — . 

Jo 4 

206 r V2 ax - x2 dx = — . 

Jo 4 

207 C ^1 u = — ^- (a and b pos.) 
Jo ax2 + b 2 Vab 

208 I „ , , = , , , = — ;=:. (a and b pos.) 
Jo ax2 + b J ji ax2 + b 4 Vab 

J"^ /-Z I . '2 . e . (n — I ) IT 
sin^axdx = | cos^axdx = — ^ j • - • v / — _ (n,evenintegO 
Jo 2«4'6'...n2a 

210 r sin'^ ax dx = j cos" ax dx = '^JiAl — * ' ' ' '-^ — ^1 -, (n, odd integ.) 
Jo Jo I • 3 • 5 • . . . n a 

sin ax sin bx dx = | cos ax cos bx dx = 0. (a 5^ b) 

Jo 

/»7r /»7r <jr 

212 I sin2axdx= ) cos2axdx=-' 

Jo Jo 2 

213 re-^dx = l\/^. 
Jo 2 T a 

214 Px^e-'^^^dx = ^. (n integ.) 



Hyperbolic Functions 15 

HYPERBOLIC FUNCTIONS 
Definitions of Hyperbolic Functions 

I 



Hyperbolic sine (sinh) x = ^ (e^ — e~^) ; csch x 



sinhx 

I 



Hyperbolic cosine (cosh) x = ^ {e^ -\- e~^) ; sech x = — ^ 

HyperboUc tangent (tanh) x = ^x^-^-x ; coth x = ^^^^^ 

where e = base of natural logarithms. 

Inverse or Anti-Hyperbolic Functions 
If X = sinh y, then y is the anti-hyperbolic sine of x or y = sinh"'^ x. 

sinh~i X = In (x + Vx^ + i) ; csch~i x = sinh~^ -• 

cosh-i X = In (x + Vx2 — i) ; sech'^ x = cosh-^ -• 

tanh~^ X = - In ; coth"^ x = tanh"^ - • 

2 I — X X 

Derivatives of Hyperbolic Functions 

3— sinh X = cosh x : -5— cosh x = sinh x ; 3- tanh x = sech^ x. 

dx dx dx 

3-cothx= — csch^x; 3-sechx= — sech x tanh x; :j-cschx= — cschxcothx. 
dx ' dx dx 

sinh~i X = , : -z- cosh~i x = , ; -r- tanh~^ x = 



dx Vx2+i ' dx Vx2 — I ' dx I — x' 

d , t- 1 id,, I d , , 

coth~i X = — r ; 3- sech~i x = — — . : 3- csch"^ x = 



dx x2-i'dx xVi-x2'dx xVx^+i 

Some Integrals Leading to Hyperbolic Functions 
I sinh X dx = cosh x ; I cosh x dx = sinh x ; j tanh x dx = In cosh x. 

J coth X dx = In sinh x ; j sech x dx = sin~i (tanh x) ; j csch x dx = In tanh - • 

/ , =sinh-i - ; f , =cosh-i -; f „ '^ „ = - tanh-^ - • (x < a) 

vx^+a^ 3. J Vx^— a^ ^ ^ a^— x2 a a 

, = — sinh~i - ; \ — ^ = — cosh~i - ; 

X Va2 + x2 a X J ^ Va^ -^ a x 

f-r^ = - - tanh-i - • (x > a) 
J x2 - a2 a X ' 

rVx2 - a2 dx = - \/x2 - a2 - - cosh-i - • 
^ 2 2 a 

rVx2 + a2 dx = - Vx2 + a2 + - sinh-i - • 
»/ 2 2 a 



16 



Natural Logarithms 



0-509 



N 





I 


2 


3 


4 


5 


6 


7 


8 


9 







. 0000 


0.6931 


1.0986 


1-3863 


1.6094 


I. 7918 


1-9459 


2.0794 


2.1972 


I 

2 

3 
4 
5 
6 

7 
8 

9 


2.3026 

9957 
3.4012 

6889 

9120 

4-0943 

2485 
3820 
4998 


2.3979 

3 -0445 

4340 

7136 

9318 

4-II09 

2627 

3944 
5109 


2.4849 
3.0910 

4657 

■ 7377 

9512 

4.1271 

2767 
4067 
5218 


2-5649 
3.1355 
4965 
7612 
9703 
4-1431 
2905 
4188 
5326 


2.6391 

3.1781 

5264 

7842 

9890 

4-1589 

3041 
4308 
5433 


2 .7081 
3-2189 

5553 

8067 

4-0073 

1744 

3175 
4427 

5539 


2.7726 
3.2581 
5835 
8286 
4-0254 
1897 
3307 
4543 
5643 


2.8332 

3.2958 

6109 

8501 

4.0431 
2047 

3438 
4659 
5747 


2 . 8904 
3.3322 
6376 
8712 
4.0604 
2195 
3567 
4773 
5850 


2.9444 

3-3673 

6636 

8918 

4-0775 

2341 

3694 
4886 

5951 


10 


6052 


6151 


6250 


6347 


6444 


6540 


6634 


6728 


6821 


6913 


II 

12 

13 
14 
IS 
i6 

17 
i8 

19 


7005 
7875 
8675 
9416 
5.0106 
0752 

1358 
1930 
2470 


7095 
7958 
8752 
9488 

5 -0173 
0814 

1417 
1985 
2523 


7185 
8040 
8828 

9558 

5-0239 

0876 

1475 
2040 

2575 


7274 
8122 

8903 

9628 

5-0304 

0938 

1533 
2095 
2627 


7362 
8203 
8978 
9698 
5-0370 
0999 

1591 
2149 

2679 


7449 
•8283 
9053 
9767 
5-0434 
1059 
1648 
2204 
2730 


7536 
8363 
9127 

9836 

5-0499 

1 1 20 

1705 
2257 
2781 


7622 
8442 
9200 

9904 
5.0562 
1180 
1761 
2311 
2832 


7707 
8520 

9273 
9972 
5.0626 
1240 
1818 
2364 
2883 


7791 
8598 
9345 
5-0039 
0689 
1299 

1^74 
2417 

2933 


20 


2983 


3033 


3083 


3132 


3181 


3230 


' 3279 


3327 


3375 


3423 


21 
22 
23 
24 

25 
26 

27 

28 
29 


3471 
3936 
4381 
4806 
5215 
5607 

5984 
6348 
6699 


3519 
3982 
4424 
4848 
5255 
5645 
6021 
6384 
6733 


3566 
4027 
4467 
4889 
5294 
5683 
6058 
6419 
6768 


3613 
4072 
4510 

4931 

5334 
5722 

6095 
6454 
6802 


3660 
4116 
4553 
4972 
5373 
5759 
6131 
6490 
6836 


3706 
4161 
4596 

5013 
5413 
5797 
6168 

6525 
6870 


3753 
4205 
4638 

5053 
5452 
5835 
6204 
6560 
6904 


3799 
4250 
4681 

5094 
5491 
5872 
6240 
6595 
6937 


3845 
4293 
4723 

5134 
5530 
5910 
6276 
6630 
6971 


3891 
4337 
4765 

5175 
5568 

5947 
6312 
6664 
7004 


30 


7038 


7071 


7104 


7137 


7170 


7203 


7236 


7268 


7301 


7333 


31 
32 
33 
34 
35 
36 

37 
38 
39 


7366 
7683 
7991 
8289 

8579 
8861 

9135 
9402 
9661 


7398 
7714 
8021 

8319 
8608 
8889 
9162 
9428 
9687 


7430 
7746 
8051 

8348 
8636 
8916 
9189 
9454 
9713 


7462 

7777 
8081 

8377 
8665 

8944 
9216 
9480 
9738 


7494 
7807 
8111 

8406 
8693 
8972 

9243 
9506 

9764 


7526 
7838 
8141 

8435 
8721 

8999 
9269 
9532 
9789 


7557 
7869 
8171 
8464 

8749 
9026 

9296 
9558 
9814 


7589 
7900 
8201 

8493 
8777 
9054 
9322 
9584 
9839 


7621 
7930 
8230 

8522 
8805 
9081 

9349 
9610 

9865 


7652 
7961 
8260 

8551 
8833 
9108 

9375 
9636 
9890 


40 


9915 


9940 


9965 


9989 


6.0014 


6.0039 


6 . 0064 


6.0088 


6.0113 


6.0137 


41 
42 

43 
44 
45 
46 

47 
48 
49 


6.0162 
0403 
0638 
0868 
1092 
1312 

1527 
1738 
1944 


6.0186 
0426 
0661 
0890 
1115 
1334 
1549 
1759 
1964 


6.0210 
0450 
0684 

0913 
1137 
1356 

1570 
1779 
1985 


6.0234 

0474 
0707 

0936 
1159 
1377 

1591 
1800 
2005 


0259 

0497 
0730 

0958 
1181 

1399 
1612 
1821 
2025 


0283 
0521 
0753 
0981 
1203 
1420 

1633 
1841 
2046 


0307 
0544 
0776 

1003 
1225 
1442 

1654 
1862 
2066 


0331 
0568 

0799 
1026 
1247 
1463 

1675 
1883 
2086 


0355 
0591 
0822 

1048 
1269 
1485 
1696 

1903 
2106 


0379 
0615 
0845 
1070 
1291 
1506 
1717 
1924 
2126 


50 


2146 


2166 


2186 


2206 


2226 


2246 


2265 


2285 


2305 


2324 


N 





I 


2 


3 


4 


s 


6 


7 


8 


9 



Natural Logarithms 



17 
500-1009 



N 





I 


2 


3 


4 


5 


6 


7 


8 


9 


50 


6.2146 


6.2166 


6.2186 


6.2206 


6.2226 


6.2246 


6.2265 


6.2285 


6.2305 


6.2324 


51 

52 

53 
54 
55 
56 

57 
58 
59 


2344 
2538 
2729 

2916 
3099 
3279 
3456 
3630 
3801 


2364 
2558 
2748 

2934 
3117 
3297 

3474 
3648 
3818 


2383 
2577 
2766 

2953 
3135 
3315 

3491 
3665 
3835 


2403 
2596 
2785 
2971 
3154 
3333 

3509 
3682 

3852 


2422 
2615 
2804 

2989 
3172 

3351 

3526 
3699 
3869 


2442 
2634 
2823 

3008 
3190 
3368 

3544 
3716 
3886 


2461 

2653 
2841 
3026 
3208 
3386 
3561 
3733 
3902 


2480 
2672 
2860 

3044 
3226 

3404 
3578 
3750 
3919 


2500 
2691 
3879 
3063 
3244 
3421 

3596 
3767 
3936 


2519 
2710 

2897 
3081 
3261 
3439 
3613 
3784 
3953 


60 


3969 


3986 


4003 


4019 


4036 


4052 


4069 


4085 


4102 


4118 


6i 
62 
63 
64 
65 
66 

67 
68 
69 


4135 
4297 
4457 

4615 

4770 
4922 

5073 
5221 

5367 


4151 
4313 
4473 
4630 
4785 
4938 
5088 
5236 
5381 


4167 
4329 
4489 
4646 
4800 
4953 
5103 
5250 
5396 


4184 
4345 
4505 
4661 
4816 
4968 

5117 
5265 
5410 


. 4200 
4362 
4520 
4677 
4831 
4983 
5132 
5280 
5425 


4216 
4378 
4536 

4693 
4846 

4998 
5147 
5294 
5439 


4232 
4394 
4552 
4708 
4862 
5013 
5162 
5309 
5453 


4249 
4409 
4568 

4723 
4877 
5028 

5177 
5323 
5468 


4265 
4425 
4583 

4739 
4892 

5043 

5191 
5338 
5482 


4281 
4441 
4599 
4754 
4907 
5058 
5206 
5352 
5497 


70 


5511 


5525 


5539 


5554 


5568 


5582 


5596 


5610 


5624 


5639 


71 

72 

73 
74 
75 
76- 

77 
78 
79 


5653 
5793 
5930 
6067 
6201 
6333 
6464 
6593 
6720 


5667 
5806 
5944 
6080 
6214 
6346 
6477 
6606 
6733 


5681 
5820 
5958 

6093 
6227 

6359 
6490 
6619 
6746 


5695 
5834 
5971 
6107 
6241 
6373 
6503 
6631 
6758 


5709 
5848 
5985 
6120 
6254 
6386 
6516 
6644 
6771 


5723 
5862 

5999 

6134 
6267 

6399 
6529 
6657 
6783 


5737 
5876 
6012 

6147 
6280 
6412 
6542 
6670 
6796 


5751 
5889 
6026 

6161 
6294 
6425 

6554 
6682 
6809 


5765 
5903 
6039 

6174 
6307 
6438 
6567 
6695 
6821 


5779 
5917 
6053 
6187 
6320 
6451 
6580 
6708 
6834 


80 


; 6846 


6859 


6871 


6884 


6896 


6908 


6921 


6933 


6946 


6958 


81 
82 
83 
84 
S5 
86 

87 
88 

89 


6970 

7093 
7214 

7334 
7452 
7569 
7685 
7799 
7912 


6983 
7105 
7226 

7346 
7464 
7581 

7696 
7811 
7923 


6995 
7117 
7238 

7358 
7476 
7593 
7708 
7822 
7935 


7007 
7130 
7250 

7370 
7488 
7604 

7719 
7833 
7946 


7020 
7142 
7262 
7382 

7499 
7616 

7731 
7845 
7957 


7032 
7154 
7274 
7393 
7511 
7627 

7742 
7856 
7968 


7044 
7166 
7286 

7405 
7523 
7639 
7754 
7867 
7979 


7056 
7178 
7298 

7417 
7534 
7650 

7765 
7878 
7991 


7069 
7190 
7310 

7429 
7546 
7662 

7776 
7890 
8002 


7081 
7202 
7322 

7441 
7558 
7673 
7788 
7901 
8013 


90 


8024 


8035 


8046 


8057 


8068 


8079 


8090 


-8101 


8112 


8123 


91 
92 
93 
94 
95 
96 

97 
98 

99 


8134 
8244 
8352 

8459 
8565 
8669 

8773 
8876 

8977 


8145 
8255 
8363 
8469 

8575 
8680 

8783 
8886 
8987 


8156 
8265 
8373 
8480 
8586 
8690 

8794 
8896 

8997 


8167 
8276 
8384 
8491 
8596 
8701 

8804 
8906 
9007 


8178 
8287 
8395 
8501 
8607 
8711 

8814 
8916 
9017 


8189 
8298 
8403 
8512 
8617 
8721 
8824 
8926 
9027 


8200 
8309 
8416 

8522 
8628 
8732 

883s 
8937 
9037 


8211 
8320 
8427 

8533 
8638 
8742 

8845 
8947 
9048 


8222 
8330 
8437 

8544 
8648 
8752 

8855 
8957 
9058 


8233 
8341 
8448 

8554 
8659 
8763 
8865 
8967 
9068 


100 


9078 


9088 


9098 


9108 


9117 


9127 


9137 


9147 


9157 


9167 


N 





I 


2 


3 


4 


5 


6 


7 


8 


9 



18 










Degrees to Radians 










Degs. 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 





0.0000 


0.0017 


0.0035 


0.0052 


0.0070 


0.0087 


0.0105 


0.0122 


0.0140 


0.0157 


I 

2 

3 

4 

5 
6 

7 
8 

9 


0.0175 
0.0349 
0.0524 
0.0698 
0.0873 
0.1047 
0.1222 
0.1396 
0.1571 


0.0192 
0.0367 
0.0541 
0.0716 
0.0890 
0.1065 
0.1239 
0.1414 
0.1588 


0.0209 
0.0384 
0.0559 

0.0733 
. 0908 
0.1082 
0.1257 
0.1431 
0.1606 


0.0227 
0.0401 
0.0576 
0.0750 
0.0925 
0,1100 
0.1274 

0.1449 
0.1623 


0.0244 
0.0419 
0.0593 
0.0768 
0.0942 
0.1117 
0.1292 
0.1466 
0.1 641 


0.0262 
0.0436 
0.0611 
0.0785 
0.0960 
0.1134 

0.1309 
0.1484 
0.1658 


0.0279 
0.0454 
0.0628 

0.0803 
0.0977 
0.1152 
0.1326 
0.1501 
0.1676 


0.0297 
0.0471 
0.0646 
0.0820 
0.0995 
0.1169 

0.1344 
0.1518 
0.1693 


0.0314 
0.0489 
0.0663 
0.0838 
0.1012 
0.1187 
0.1361 
0.1536 
0.1710 


0.0332 
0.0506 
0.0681 
0.0855 
. 1030 
0.1204 

0.1379 
0.1553 
0.1728 


10 


0.1745 


0.1763 


0.1780 


0.1798 


0.1815 


0.1833 


0.1850 


0.1868 


0.1885 


0.1902 


II 

12 

13 
14 
15 
i6 

17 
i8 

19 


0.1920 
. 2094 
0.2269 
0.2443 
0.2618 
0.2793 
0.2967 
0.3142 
0.3316 


0.1937 
0.2112 
0.2286 
0.2461 
0.2635 
0.2810 
0.2985 
0.3159 
0.3334 


0.1955 
0.2129 
. 2304 
0.2478 
0.2653 
0.2827 
0.3002 
0.3176 
0.3351 


0.1972 
0.2147 
0.2321 
0.2496 
0.2670 
. 2845 
0.3019 
0.3194 
0.3368 


0.1990 
0.2164 
0.2339 
0.2513 
0.2688 
0.2862 

0.3037 
0.3211 
0.3386 


0.2007 
0.2182 
0.2356 
0.2531 
0.2705 
. 2880 

0.3054 
0.3229 
0.3403 


0.2025 
0.2199 
0.2374 
0.2548 
0.2723 
0.2897 
0.3072 
0.3246 
0.3421 


0.2042 
0.2217 
0.2391 
0.2566 
0.2740 
0.2915 

0.3089 
0.3264 
0.3438 


0.2059 
0.2234 
. 2409 
0.2583 
0.2758 
0.2932 
0.3107 
0.3281 
0.3456 


0.2077 
0.2251 
0.2426 
0.2601 
0.2775 
0.2950 
0.3124 
0.3299 
0.3473 


20 


0.3491 


0.3508 


0.3526 


0.3543 


0.3560 


0.3578 


0.3595 


0.3613 


0.3630 


0.3648 


21 
22 
23 
24 
25 
26 

27 
28 

29 


0.3665 
0.3840 
0.4014 
0.4189 
0.4363 
0.4538 
0.4712 
0.4887 
0.5061 


0.3683 

0.3857 
0.4032 

0.4206 
0.4381 
0.4555 
0.4730 
0.4904 
0.5079 


0.3700 
0.3875 
0.4049 
0.4224 
0.4398 
0.4573 
0.4747 
0.4922 
. 5096 


0.3718 
0.3892 
0.4067 
0.4241 
0.4416 
0.4590 

0.4765 
0.4939 
0.5114 


0.3735 
0.3910 
. 4084 
0.4259 

0.4433 
0.4608 

0.4782 
0.4957 
0.5131 


0.3752 
0.3927 
0.4102 
0.4276 

0.4451 
0.4625 

0.4800 
0.4974 
0.5149 


0.3770 
0.3944 
0.4119 

0.4294 
0.4468 
0,4643 
0.4817 
0.4992 
0.5166 


0.3787 
0.3962 
0.4136 
0.4311 
0.4485 
0.4660 

0.4835 
0.5009 
0.5184 


0.3805 
0.3979 
0.4154 
0.4328 
0.4503 
0.4677 
0.4852 
0.5027 
0.5201 


0.3822 

0.3997 
0.4171 

0.4346 
0.4520 
0.4695 
0.4869 
0.5044 
0.5219 


30 


0.5236 


0.5253 


0.5271 


0.5288 


0.5306 


0.5323 


0.5341 


0.5358 


0.5376 


0.5393 


31 

32 

3S 
34 
35 
36 

37 
38 
39 


0.5411 

0.5585 
0.5760 

0.5934 
0.6109 
0.6283 
0.6458 
0.6632 
0.6807 


0.5428 
0.5603 
0.5777 
0.5952 
0.6126 
0.6301 

0.6475 
0.6650 
0.6824 


0.5445 
0.5620 

0.5794 

0.5969 
0.6144 
0.6318 

0.6493 
0.6667 
. 6842 


0.5463 
0.5637 
0.5812 

0.5986 
0.6161 
0.6336 
0.6510 
0.6685 
0.6859 


. 5480 

0.5655 
0.5829 

. 6004 
0.6178 
0.6353 
0.6528 
0.6702 
0.6877 


0.5498 
0.5672 
0.5847 
0.6021 
0.6196 
0.6370 

0.6545 
0.6720 
0.6894 


0.5515 
0.5690 
0.5864 
0.6039 
0.6213 
0.6388 
0.6562 
0.6737 
0.6912 


0.5533 
0.5707 
0.5882 
. 6056 
0.6231 
. 6405 
0.6580 

0.6754 
0.6929 


0.5550 
0.5725 
0.5899 
. 6074 
0.6248 
0.6423 
0.6597 
0.6772 
. 6946 


0.5568 
0.5742 
0.5917 
0.6091 
0.6266 
.6440 
0.6615 
0.6789 
. 6964 


40 


0.6981 


0.6999 


o.70'i6 


. 7034 


0.7051 


. 7069 


. 7086 


0.7103 


0.7121 


0.7138 


41 
42 

43 
44 


0.7156 
0.7330 
0.7505 
0.7679 


0.7173 
0.7348 
0.7522 
0.7697 


0.7191 
0.7365 
0.7540 
0.7714 


0.7208 
0.7383 
0.7557 
0.7732 


0.7226 
. 7400 
0.7575 
0.7749 


0.7243 
0.7418 
0.7592 
0.7767 


0.7261 

0.7435 
0.7610 

0.7784 


0.7278 

0.7453 
0.7627 
. 7802 


0.7295 
0.7470 
. 7645 
0.7819 


0.7313 
0.7487 
0.7662 
0.7837 


45 


0.7854 


0.7871 


0.7889 


. 7906 


0.7924 


0.7941 


0.7959 


0.7976 


0.7994 


0.8011 




0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


90° = 1.5 708 radians 
i8o° = 3.i4i6 radians 
27o°=4.7i24 radians 
360° = 6. 2832 radians 


30°=^, 45°=;. 60°=^, 90° = ; radians 

D 4 3 2 

120°= — , 135°= — . 150°=^, 180°= TT radians 
340 

210°=^, 225°=^, 240°=^, 27o°=^radians 
043 2 

300°=^, 315°= V' 330°=^, 36o° = 27r radians 













Degrees to Rj 


idians 








19 


Degs. 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


45 


0.7854 


0.7871 


0.7889 


. 7906 


0.7924 


- 7941 


0.7959 


0.7976 


0.7994 


0.801 I 


46 


0.8029 


0.8046 


0.8063 


0.8081 


0.8098 


0.8116 


0.8133 


0.8151 


0.8168 


0.8186 


47 


0.8203 


0.8221 


0.8238 


0.8255 


0.8273 


0.8290 


0.8308 


0.8325 


0.8343 


0.8360 


48 


0.8378 


0.8395 


0.8412 


0.8430 


0.8447 


0.8465 


0.8482 


. 8500 


0.8517 


0.8535 


49 


0.8552 


0.8570 


0.8587 


0.8604 


0.8622 


0.8639 


0.8657 


0.8674 


0.8692 


0.8709 


50 


0.8727 


0.8744 


0.8762 


0.8779 


0.8796 


0.8814 


0.8831 


0.8849 


0.8866 


0.8884 


51 


0.8901 


0.8919 


0.8936 


0.8954 


0.8971 


0.8988 


. 9006 


0.9023 


0.9041 


0.9058 


52 


0.9076 


0.9093 


0.9111 


0.9128 


0.9146 


0.9163 


0.9180 


0.9198 


0.9215 


0.9233 


53 


0.9250 


0.9268 


0.9285 


0.9303 


0.9320 


0.9338 


0.9355 


0.9372 


0.9390 


0.9407 


54 


0.9425 


0.9442 


. 9460 


0.9477 


0.9495 


0.9512 


0.9529 


0-9547 


0.9564 


0.9582 


55 


0.9599 


0.9617 


0.9634 


0.9652 


0.9669 


0.9687 


0.9704 


0.9721 


0.9739 


0.9756 


56 


0.9774 


0.9791 


0.9809 


0.9826 


0.9844 


0.9861 


0.98790.9896 


0.9913 


0.9931 


57 


0.9948 


0.9966 


0.9983 


I. 0001 


I .0018 


1.0036 


1-0053 


I .0071 


1.0088 


I. 0105 


58 


I. 0123 


I .0140 


I. 0158 


I. 0175 


I -0193 


I .0210 


I .0228 


1.0245 


1.0263 


I .0280 


59 


1.0297 


I -0315 


1.0332 


1.0350 


1.0367 


I -0385 


I .0402 


1 .0420 


1-0437 


1-0455 


60 


1.0472 


1 .0489 


1.0507 


1.0524 


I .0542 


1-0559 


1-0577 


1.0594 


I .0612 


I .0629 


61 


I .0647 


I .0664 


I. 0681 


I .0699 


I .0716 


1.0734 


I. 0751 


I .0769 


1.0786 


I .0804 


62 


I. 0821 


1.0838 


I .0856 


1.0873 


I .0891 


I .0908 


I .0926 


1.0943 


I .0961 


I .0978 


63 


1.0996 


1.1013 


I. 1030 


I . 1048 


I. 1065 


I. 1083 


I .1100 


1.1118 


1.1135 


I .1153 


64 


I .1170 


1.1188 


I. 1205 


I .1222 


I .1240 


I. 1257 


1. 1275 


I .1292 


1.1310 


1. 1327 


65 


I -1345 


I. 1362 


I. 1380 


1. 1397 


1.1414 


I. 1432 


I. 1449 


I. 1467 


I . 1484 


1 .1502 


66 


11519 


I-I537 


I -1554 


1. 1572 


I. 1589 


I. 1606 


I .1624 


1.1641 


1. 1659 


I. 1676 


67 


I. 1694 


1 .1711 


1. 1729 


1. 1746 


I. 1764 


1.1781 


I. 1798 


1.1816 


1. 1833 


1.1851 


68 


I. 1868 


I. 1886 


1. 1903 


1.1921 


I -1938 


1. 1956 


I -1973 


I. 1990 


I .2008 


1.2025 


69 


I . 2043 


I . 2060 


I .2078 


1.2095 


1.2113 


1 .2130 


I. 2147 


1-2165 


1. 2182 


I .2200 


70 


1 .2217 


1-2235 


1.2252 


I .2270 


1.2287 


1-2305 


1.2322 


1-2339 


1.2357 


1.2374 


71 


1.2392 


1.2409 


1.2427 


I . 2444 


I .2462 


1.2479 


1.2497 


1. 2514 


1. 2531 


1.2549 


72 


1.2566 


1-2584 


I .2601 


1,2619 


1.2636 


1.2654 


I .2671 


1.2689 


I .2706 


1.2723 


73 


1-2741 


1.2758 


1.2776 


1.2793 


1.2811 


1.2828 


1.2846 


1.2863 


I. 2881 


1.2898 


74 


1.2915 


1-2933 


1.2950 


1.2968 


1.2985 


1.3003 


1.3020 


1-3038 


1-3055 


1.3073 


75 


1.3090 


1-3107 


1-3125 


I. 3142 


I. 3160 


1-3177 


1. 3195 


1. 3212 


1.3230 


1-3247 


76 


1-3265 


1.3282 


1-3299 


1-3317 


1-3334 


1-3352 


1.3369 


1-3387 


1.3404 


1.3422 


77 


1-3439 


1-3456 


1.3474 


I -3491 


1-3509 


1-3526 


1-3544 


I -3561 


1.3579 


1.3596 


78 


1-3614 


I -3631 


1.3648 


1.3666 


1.3683 


I. 3701 


1. 3718 


1-3736 


1-3753 


1.3771 


79 


1.3788 


1.3806 


1.3823 


1.3840 


1-3858 


1-3875 


1-3893 


I -3910 


1.3928 


1-3945 


80 


I -3963 


1-3980 


1.3998 


I. 4015 


1.4032 


I .4050 


1.4067 


1.4085 


I. 4102 


I. 4120 


81 


I -4137 


I -4155 


I. 4172 


I. 4190 


1.4207 


1.4224 


1.4242 


1-4259 


1.4277 


I .4294 


82 


I. 4312 


1-4329 


1-4347 


1.4364 


1.4382 


1-4399 


I. 4416 


1-4434 


1.4451 


1.4469 


83 


I .4486 


1-4504 


1.4521 


1-4539 


1.4556 


1.4573 


I -4591 


1.4608 


1.4626 


1-4643 


84 


I .4661 


1.4678 


1.4696 


I -4713 


I. 4731 


1.4748 


1.4765 


1-4783 


I . 4800 


I. 4818 


85 


1-4835 


1-4853 


I .4870 


1.4888 


1.4905 


1.4923 


1.4940 


1-4957 


1-4975 


1.4992 


86 


I. 5010 


1.5027 


1-5045 


1.5062 


I . 5080 


1.5097 


1-5115 


I. 5132 


I. 5149 


I -5167 


87 


I. 5184 


1 .5202 


1-5219 


1-5237 


1.5254 


1.5272 


1-5289 


1-5307 


1.5324 


I -5341 


88 


1-5359 


1-5376 


1-5394 


1.5411 


1-5429 


1.5446 


1.5464 


I. 5481 


1.5499 


I -5516 


89 


1-5533 


I -5551 


1-5568 


1.5586 


1-5603 


I .5621 


1-5638 


1.5656 


1-5673 


1. 5691 


90 


1.5708 


1-5725 


1-5743 


1.5760 


1-5778 


1-5795 


I -5813 


1.5830 


1-5848 


1.5865 




0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 


90° = 


=1.5708 radians 


30° = 5, 45°=-, 60°=-, 90°= -radians 
043 2 


180° = 


= 3.1416 radians 


120°= — , 135°=^, 150°=?, 180°= TT radians 


270^ = 


= 4.7124 radians 


210°=^, 225°=^, 240°=^, 27o°=^radians 
643 2 


360° = 


= 6.2832 radians 


300°=^, 315°=^, 330°= i^,36o° = 27r radians 

3 46 



20 



Natural Sines, Cosines and Tangents 























0^ 


-14.9" 


Degs. 


Function 


o.o" 


0.1° 


0.2° 


0.3° 


0.4° 


0.5° 


0.6° 


0.7° 


0.8" 


0.9° 




sin 


o.oooo 


0.0017 


0.0035 


0.0052 


0.0070 


0.0087 


0.0105 


0.0122 


0.0140 


0.0157 





cos 


I . 0000 


I. 0000 


I . 0000 


I. 0000 


I. 0000 


I . 0000 


0.9999 


0.9999 


0.9999 


0.9999 




tan 


0,0000 


0.0017 


0.0035 


0.0052 


0.0070 


0.0087 


0.0105 


0.0122 


0.0140 


0.0157 




sin 


0.0175 


0.0192 


0.0209 


0.0227 


0.0244 


0.0262 


0.0279 


0.0297 


0.0314 


0.0332 


1 


cos 


0.9998 


0.9998 


0.9998 


0.9997 


0.9997 


0.9997 


0.9996 


0.9996 


0.999s 


0.9995 




tan 


0.017s 


0.0192 


0.0209 


0.0227 


0.0244 


0.0262 


0.0279 


0.0297 


0.0314 


0.0332 




sin 


0.0349 


0.0366 


0.0384 


0.0401 


0.0419 


0.0436 


0.0454 


0.0471 


0.0488 


0.0506 


2 


cos 


0.9994 


0.9993 


0.9993 


0.9992 


0.9991 


0.9990 


0.9990 


0.9989 


0.9988 


0.9987 




tan 


0.0349 


0.0367 


0.0384 


0.0402 


0.0419 


0.0437 


0.04S4 


0.0472 


0.0489 


0.0507 




sin 


0.0523 


0.0541 


0.0558 


0.0576 


0.0593 


0.0610 


0.0628 


0.0645 


0.0663 


0.0680 


3 


cos 


0.9986 


0.9985 


0.9984 


0.9983 


0.9982 


0.9981 


0.9980 


0.9979 


0.9978 


0.9977 




tan 


0.0524 


0.0542 


0.0559 


0.0577 


0.0594 


0.0612 


0.0629 


0.0647 


0.0664 


0.0682 




sin 


0.0698 


0.0715 


0.0732 


0.0750 


0.0767 


0.0785 


0.0802 


0.0819 


0.0837 


0.0854 


4 


cos 


0.9976 


0.9974 


0.9973 


0.9972 


0.9971 


0.9969 


0.9968 


0.9966 


0.9965 


0.9963 




tan 


0.0699 


0.0717 


0.0734 


0.0752 


0.0769 


0.0787 


0.0805 


0.0822 


0.0840 


0.0857 




sin 


0.0872 


0.0889 


0.0906 


0.0924 


0.0941 


0.0958 


0.0976 


0.0993 


O.IOII 


0.1028 


5 


cos 


0.9962 


0.9960 


0.9959 


0.9957 


0.9956 


0.9954 


0.9952 


0.9951 


0.9949 


0.9947 




tan 


0.0875 


0.0892 


0.0910 


0.0928 


0.0945 


0.0963 


0.0981 


0.0998 


O.IOI6 


0.1033 




sin 


0.1045 


0.1063 


0.1080 


0.1097 


o.iiis 


0.1132 


O.I149 


0.1167 


O.II84 


0.1201 


6 


cos 


0.9945 


0.9943 


0.9942 


0.9940 


0.9938 


0.9936 


0.9934 


0.9932 


0.9930 


0.9928 




tan 


0.1051 


0.1069 


0.1086 


0.1104 


0.1122 


0.1139 


0.IIS7 


0.1175 


O.II92 


0.1210 




sin 


0.1219 


0.1236 


O.I2S3 


0.1271 


0.1288 


0.1305 


0.1323 


0.1340 


0.1357 


0.1374 


7 


cos 


0.9925 


0.9923 


0.9921 


0.9919 


0.9917 


0.9914 


0.9912 


0.9910 


0.9907 


0.990s 




tan 


0.1228 


0.1246 


0.1263 


0.1281 


0.1299 


0.1317 


0.1334 


O.I3S2 


0.1370 


0.1388 




sin 


0.1392 


0.1409 


0.1426 


0.1444 


0.1461 


0.1478 


0.1495 


0.IS13 


0.1530 


0.1547 


8 


cos 


0.9903 


0.9900 


0.9898 


0.9895 


0.9893 


0.9890 


0.9888 


0.9885 


0.9882 


0.9880 




tan 


0.1405 


0.1423 


0.1441 


0.1459 


0.1477 


0.1495 


0.1512 


0.1530 


0.1548 


0.1566 




sin 


0.1564 


0.1582 


0.1599 


0.1616 


0.1633 


0.1650 


0.1668 


0.1685 


0.1702 


0.1719 


9 


cos 


0.9877 


0.9874 


0.9871 


0.9869 


0.9866 


0.9863 


0.9860 


0.9857 


0.9854 


0.9851 




tan 


0.1584 


0.1602 


0.1620 


0.1638 


0.1655 


0.1673 


0.1691 


0.1709 


0.1727 


0.174s 




sin 


1736 


0.1754 


0.1771 


0.1788 


0.1805 


0.1822 


0.1840 


0.1857 


0.1874 


0.1891 


10 


cos 


0.9848 


0.9845 


0.9842 


0.9839 


0.9836 


0.9833 


0.9829 


0.9826 


0.9823 


0,9820 




tan 


0.1763 


0.1781 


0.1799 


0.1817 


0.1835 


0.1853 


0.1871 


0.1890 


0.1008 


0.1926 




sin 


0.1908 


0.192s 


0.1942 


0.1959 


0.1977 


0.1994 


0.2011 


0.2028 


0.204s 


0,2062 


11 


cos 


0.9816 


0.9813 


0.9810 


0.9806 


0.9803 


0.9799 


0.9796 


0.9792 


0.9789 


0.9785 




tan 


0.1944 


0.1962 


0.1980 


0.1998 


0.2016 


0.2035 


0.2053 


0.2071 


0.2089 


0.2107 




sin 


0.2079 


0.2096 


0.2113 


0.2130 


0.2147 


0.2164 


0.2181 


0.2198 


0.2215 


0.2232 


12 


cos 


0.9781 


0.9778 


0.9774 


0.9770 


0.9767 


0.9763 


0.97S9 


0.975S 


0.9751 


0.9748 




tan 


0.2126 


0.2144 


0,2162 


0.2180 


0.2199 


0.2217 


0.2235 


0.22S4 


0.2272 


0.2290 




sin 


0.2250 


0.2267 


0.2284 


0.2300 


0.23x8 


0.2334 


2351 


0.2368 


0.238s 


0.2402 


13 


cos 


0.9744 


0.9740 


0.9736 


0.9732 


0.9728 


0.9724 


0.9720 


0.971S 


0.97II 


0.9707 




tan 


0.2309 


0.2327 


0.2345 


0.2364 


0.2382 


0.2401 


0.2419 


0.2438 


0.2456 


0.2475 




sin 


0.2419 


0.2436 


0.2453 


0.2470 


0.2487 


0.2504 


0.2521 


0.2538 


0.25S4 


0.2571 


14 


cos 


0.9703 


0.9699 


0.9694 


0.9690 


0.9686 


0.9681 


0.9677 


0.9673 


0.9668 


0.9664 




tan 


0.2493 


0.2512 


0.2530 


0,2549 


0.2568 


0.2586 


0.260s 


0.2623 


0.2642 


0.2661 


Degs. 


Function 


0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 



Natural Sines, Cosines and Tangents 



21 
15°-29.9° 



Degs. 


Function 


o.o° 


0.1° 


0.2" 


0.3° 


0.4° 


0.5° 


0.6' 


0.7° 


0.8° 


0.9° 




sin 


0.2588 


. 2605 


0.2622 


0.2639 


. 2656 


0.2672 


0.2689 


0.2706 


0.2723 


0.2740 


15 


cos 


0.9659 


0.9655 


0.9650 


0.9646 


0.9641 


0.9636 


0.9632 


0.9627 


0.9622 


0.9617 




tan 


0.2679 


0.2698 


0.2717 


0.2736 


0.2754 


0.2773 


0.2792 


0.2811 


0.2830 


0.2849 




sin 


0.2756 


0.2773 


0.2790 


0.2807 


0.2823 


0.2840 


0.2857 


0.2874 


0.2890 


0.2907 


16 


cos 


0.9613 


0.9608 


0.9603 


0.9598 


0.9593 


0.9588 


0.9583 


0.0578 


0.9573 


0.9568 




tan 


0.2867 


0.2886 


0.2905 


0.2924 


0.2943 


0.2962 


0.2981 


0.3000 


0.3019 


3038 




sin 


0.2924 


0.2940 


0.2957 


0.2974 


0.2990 


0.3007 


0.3024 


0.3040 


3057 


0.3074 


17 


cos 


0.9563 


0.9558 


0.95S3 


0.9548 


0.9542 


0.9537 


0.9532 


9527 


0.9521 


0.9516 




tan 


0.3057 


0.3076 


0.3096 


0.3115 


0.3134 


0.3153 


0.3172 


3191 


0.3211 


3230 




sin 


0.3090 


0.3107 


0.3123 


0.3140 


0.3156 


0.3173 


0.3190 


0.3206 


0.3223 


0.3239 


18 


cos 


0.9511 


0.9505 


0.9500 


0.9494 


0.9489 


0.9483 


0.9478 


0.9472 


0.9466 


0.9461 




tan 


0.3249 


0.3269 


0.3288 


0.3307 


0.3327 


0.3346 


0.3365 


0.3385 


0.3404 


0.3424 




sin 


0.3256 


0.3272 


0.3289 


0.330s 


0.3322 


0.3338 


0.3355 


0.3371 


0.3387 


0.3404 


19 


cos 


0.9455 


0.9449 


0.9444 


0.9438 


0.9432 


0.9426 


0.9421 


0.9415 


0.9409 


0.9403 




tan 


0.3443 


0.3463 


0.3482 


0.3502 


0.3522 


0.3541 


0.3561 


3581 


0.3600 


0.3620 




sin 


0.3420 


0.3437 


0.3453 


0.3469 


0.3486 


0.3502 


0.3518 


0.3535 


0.3551 


0.3567 


20 


cos 


0.9397 


0.9391 


0.9385 


0.9379 


0.9373 


0.9367 


0.9361 


0.9354 


0.9348 


0.9342 




tan 


0.3640 


0.3659 


0.3679 


0.3699 


0.3719 


0.3739 


0.3759 


0.3779 


0.3799 


0.3819 




sin 


0.3584 


0.3600 


0.3616 


0.3633 


0.3649 


0.3665 


0.3681 


0.3697 


0.3714 


0.3730 


21 


cos 


0.9336 


0.9330 


0.9323 


0.9317 


0.9311 


0.9304 


0.9298 


0.9291 


0.9285 


0.9278 




tan 


0.3839 


0.3859 


0.3879 


0.3899 


0.3919 


0.3939 


0.3959 


0.3979 


0.4000 


0.4020 


22 


sin 


0.3746 


0.3762 


0.3778 


0.3795 


0.3811 


0.3827 


0.3843 


0.3859 


0.3875 


0.3891 


cos 


0.9272 


0.9265 


0.9259 


0.9252 


0.9245 


0.9239 


0.9232 


0.9225 


0.9219 


0.9212 




tan 


0.4040 


0.4061 


0.4081 


0.4101 


0.4122 


0.4142 


0.4163 


0.4183 


0.4204 


0.4224 




sin 


0.3907 


0.3923 


0.3939 


0.3955 


0.3971 


0.3987 


0.4003 


0.4019 


0.4035 


0.4051 


23 


cos 


0.9205 


0.9198 


0.9191 


0.9184 


0.9178 


0.9171 


9164 


0.9157 


0.9150 


0.9143 




tan 


0.4245 


0.4265 


0.4286 


0.4307 


0.4327 


0.4348 


0.4369 


0.4390 


0.4411 


0.4431 




sin 


0.4067 


0.4083 


0.4099 


0.411S 


0.4131 


0.4147 


0.4163 


0.4179 


0.4195 


0.4210 


24 


cos 


0.9135 


0.9128 


0.9121 


0.9114 


0.9107 


0.9100 


0.9092 


0.9085 


0.9078 


0.9070 




tan 


0.4452 


0.4473 


0.4494 


0.4515 


0.4536 


0.4557 


0.4578 


0.4599 


0.4621 


0.4642 




sin 


0.4226 


0.4242 


0.4258 


0.4274 


0.4289 


0.4305 


0.4321 


0.4337 


0.4352 


0.4368 


25 


cos 


0.9063 


0.9056 


. 9048 


. 9041 


0.9033 


0.9026 


0.9018 


0.9011 


0.9003 


0.8996 




tan 


0.4663 


0.4684 


0.4706 


0.4727 


0.4748 


0.4770 


0.4791 


0.4813 


0.4834 


0.4856 




sin 


0.4384 


0.4399 


0.4415 


0.4431 


0.4446 


0.4462 


0.4478 


0.4493 


0.4509 


0.4524 


26 


cos 


0.8988 


0.8980 


0.8973 


0.8965 


0.8957 


0.8949 


0.8942 


0.8934 


0.8926 


0.8918 




tan 


0.4877 


0.4899 


0.4921 


0.4942 


0.4964 


0.4986 


0.5008 


0.5029 


0.5051 


0.5073 




sin 


0.4540 


0.4555 


0.4571 


0.4586 


0.4602 


0.4617 


0.4633 


0.4648 


0.4664 


0.4679 


27 


cos 


0.8910 


. 8902 


0.8894 


0.8886 


0.8878 


0.8870 


0.8862 


0.8854 


0.8846 


0.8838 




tan 


0.5C95 


0.5117 


0.5139 


0.5161 


0.5184 


0.5206 


0.5228 


0.5250 


0.5272 


0.5295 




sin 


0.4695 


0.4710 


0.4726 


0.4741 


0.4756 


0.4772 


0.4787 


0.4802 


0.4818 


0.4833 


28 


cos 


0.8829 


0.8821 


0.8813 


0.8805 


0.8796 


0.8788 


0.8780 


0.8771 


0.8763 


0.8755 




tan 


0.5317 


0.5340 


0.5362 


0.5384 


0.5407 


0.5430 


0.5452 


0.5475 


0.5498 


0.5520 




sin 


0.4848 


0.4863 


0.4879 


0.4894 


0.4909 


0.4924 


0.4939 


0.4955 


0.4970 


0.4985 


29 


cos 


0.8746 


0.8738 


0.8729 


0.8721 


0.8712 


0.8704 


0.8695 


0.8686 


0.8678 


0.8669 




tan 


0.5543 


0.5566 


0.5589 


0.5612 


0.5635 


0.5658 


0.5681 


0.5704 


0.5727 


0.5750 


Degs. 


Function 


0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 



22 



Natural Sines, Cosines and Tangents 























30M4.9° 


Degs. 


Function 


o.o° 


o.i" 


0.2° 


0.3° 


0.4" 


0.5° 


0.6^ 


0.7° 


0.8° 


0.9" 




sin 


0.5000 


5015 


. 5030 


0.5045 


0.5060 


0.5075 


0.5090 


0.5105 


0.5120 


0.5135 


30 


cos 


0.8660 


0.8652 


0.8643 


0.8634 


0.8625 


0.8616 


0.8607 


0.8599 


0.8590 


0.8581 




tan 


0.5774 


0.5797 


0.5820 


0.5844 


0.5867 


0,5890 


0.5914 


0.5938 


0.5961 


0.5985 




sin 


0.5150 


0.5165 


0.5184 


0.5195 


0.5210 


0.5225 


0.5240 


0.5255 


0.5270 


0.5284 


31 


cos 


0.8572 


0.8563 


0.8550 


0.8545 


0.8536 


0.8526 


0.8517 


0.8508 


0.8499 


0.8490 




tan 


0.6009 


0.6032 


0.6056 


0.6080 


0.6104 


0.6128 


0.6152 


0.6176 


0.6200 


, 6224 




sin 


0.5299 


0.5314 


0.5329 


0.5344 


0.5358 


0.5373 


0,5388 


0.5402 


0.5417 


0.5432 


32 


cos 


0.8480 


0.8471 


0.8462 


0.8453 


0.8443 


0.8434 


0.8425 


0.8415 


0.8406 


08396 




tan 


0.6249 


0.6273 


. 6297 


0.6322 


0.6346 


0.6371 


0.6395 


0.6420 


0.6445 


0,6469 




sin 


0.5446 


0.5461 


0.5476 


0.5490 


0.5505 


0.5519 


0.5534 


0.5548 


0,5563 


0.5577 


33 


cos 


0.8387 


0.8377 


0.8368 


0.8358 


0.8348 


0.8339 


0.8329 


0.8320 


0.8310 


0,8300 




tan 


0.6494 


0.6519 


0.6544 


0.6569 


0.6594 


0.6619 


0.6644 


0.6669 


0.6694 


6720 




sin 


0.5592 


0.5606 


0.5621 


0.5635 


0.5650 


0.5664 


0.5678 


0,5693 


0.5707 


0.5721 


34 


cos 


0,8290 


0.8281 


0.8271 


0.8261 


0.8251 


0.8241 


0.8231 


, 8221 


0.8211 


0.8202 




tan 


0.6745 


0.6771 


0.6796 


0.6822 


0.6847 


0.6873 


0.6899 


0.6924 


0,6950 


0.6976 




sin 


0.5736 


0.5750 


0.5764 


0.5779 


0.5793 


0.5807 


0.5821 


0.5835 


0.5850 


0.5864 


35 


cos 


0.8192 


0.8181 


0.8171 


0.8161 


0.8151 


0.8141 


0.8131 


0.8121 


0.8111 


0.8100 




tan 


0.7002 


0.7028 


0-7054 


0.7080 


0.7107 


0.7133 


0.7159 


0.7186 


0.7212 


0.7239 




sin 


0.5878 


0.5892 


0.5906 


0.5920 


0.5934 


0.5948 


0.5962 


0.5976 


0,5990 


0,6004 


36 


cos 


0.8090 


0.8080 


0.8070 


0.8059 


0.8049 


0.8039 


0.8028 


0.8018 


0.8007 


0.7997 




tan 


0.7265 


0,7292 


0.7319 


0,7346 


0.7373 


0.7400 


0.7427 


0.7454 


0.7481 


0,7508 




sin 


0.6018 


0.6032 


0.6046 


0.6060 


. 6074 


0.6088 


0.6101 


0.611S 


0.6129 


0.6143 


37 


cos 


0.7986 


0.7976 


0.7965 


0.7955 


0.7944 


0.7934 


0.7923 


0.7912 


0.7902 


0.7891 




tan 


0.7536 


0.7563 


0.7590 


0.7618 


0.7646 


0.7673 


0.7701 


0.7729 


0.7757 


7785 




sin 


0.6157 


0,6170 


0.6184 


0.6198 


0.6211 


0.6225 


0.6239 


0.6252 


0.6266 


0.6280 


38 


cos 


0.7880 


0.7869 


0.7859 


0.7848 


0.7837 


0.7826 


0.7815 


0.7804 


0,7793 


0.7782 




tan 


0.7813 


0.7841 


0.7869 


0.7898 


0.7926 


0.7954 


0.7983 


0.8012 


0,8040 


0.8069 




sin 


0.6293 


0.6307 


0.6320 


0.6334 


0,6347 


0.6361 


0.6374 


0.6388 


0.6401 


0.6414 


39 


cos 


0.7771 


0.7760 


0.7749 


0.7738 


0.7727 


0.7716 


0.7705 


0.7694 


0.7683 


0.7672 




tan 


0.8098 


0.8127 


8156 


0.8185 


0.8214 


0.8243 


0.8273 


0.8302 


0.8332 


0.8361 




sin 


0.6428 


0.6441 


0.6455 


0.6468 


0.6481 


0,6494 


0.6508 


0.6521 


0.6534 


0.6547 


40 


cos 


0.7660 


0.7649 


0.7638 


0.7627 


0.7615 


0.7604 


0.7593 


0.7581 


0.7570 


0.7559 




tan 


0.8391 


0.8421 


0.8451 


0.8481 


0.8511 


0.8541 


0.8571 


0.8601 


8632 


0.8662 




sin 


0.6561 


0.6574 


0.6587 


0.6600 


0,6613 


0.6626 


0.6639 


0.6652 


0.6665 


0.6678 


41 


. cos 


0.7547 


0.7536 


0.7524 


0.7513 


0.7501 


0.7490 


0.7478 


0.7466 


0.7455 


0.7443 




tan 


0.8693 


0.8724 


0.8754 


0.8785 


0.8816 


0.8847 


0.8878 


0.8910 


0.8941 


0,8972 




sin 


0.6691 


0.6704 


0.6717 


0.6730 


0.6743 


0.6756 


0.6769 


0.6782 


0.6794 


0.6807 


42 


cos 


0.7431 


0.7420 


0.7408 


0.7396 


0.7385 


0.7373 


0.7361 


0.7349 


0.7337 


7325 




tan 


0.9004 


0.9036 


0.9067 


0.9099 


0.9131 


0.9163 


0.9195 


0.9228 


0.9260 


0.9293 




sin 


0.6820 


0.6833 


0.6845 


0.6858 


0.6871 


0.6884 


0.6896 


0.6909 


0.6921 


0,6934 


43 


cos 


0.7314 


0.7302 


0.7290 


0.7278 


0.7266 


0.7254 


0.7242 


0.7230 


0.7218 


0.7206 




tan 


0.9325 


0.9358 


0.9391 


0.9424 


0.9457 


0.9490 


0.9523 


0.9556 


0.9590 


0.9623 




sin 


0.6947 


0.6959 


0.6972 


0.6984 


0.6997 


0.7009 


0.7022 


0.7034 


0.7046 


0.7059 


44 


cos 


0.7193 


0.7181 


0.7169 


0.7157 


0.7145 


0.7133 


0.7120 


7108 


0.7096 


0.7083 




tan 


0.9657 


0.9691 


0.9725 


0.9759 


0.9793 


0.9827 


0.9861 


0.9896 


0.9930 


0.9965 


Degs. 


Function 


0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 



Natural Sines, Cosines and Tangents 



23 



Degs. 


Function 


o.o° 


o.i^ 


0.2° 


0.3° 


0.4° 


0.5° 


0.6° 


0.7° 


0.8° 


0.9° 




sin 


0.7071 


0.7083 


0.7096 


0.7108 


0.7120 


0.7133 


0.7145 


0.7157 


0.7169 


0.7181 


45 


cos 


0.7071 


0.7059 


. 7046 


0.7034 


0.7022 


0.7009 


0.6997 


0.6984 


0.6972 


0.6959 




tan 


I. 0000 


1.003s 


1.0070 


I. 0105 


1.0141 


I. 0176 


I. 0212 


1.0247 


1.0283 


I. 0319 




sin 


0.7193 


0.7206 


0.7218 


0.7230 


0.7242 


0.7254 


0.7266 


0.7278 


0.7290 


0.7302 


46 


cos 


0.6947 


0.6934 


0.6921 


0.6909 


0.6896 


0.6884 


0.6871 


0.6858 


0.6845 


0.6833 




tan 


I 0355 


1.0392 


1.0428 


1.0464 


I. 0501 


1.0538 


I . 0575 


I. 0612 


1.0649 


1.0686 




sin 


0.7314 


0.7325 


0.7337 


0.7349 


0.7361 


0.7373 


0.738s 


0.7396 


0.7408 


0.7420 


47 


cos 


0.6820 


0.6807 


0.6794 


0.6782 


0.6769 


. 6756 


0.6743 


0.6730 


0.6717 


0.6704 




tan 


1.0724 


I . 0761 


1.0799 


1.0837 


1.0875 


I. 0913 


I. 0951 


1.0990 


I . 1028 


I . 1067 




sin 


0.7431 


0.7443 


0.7455 


0.7466 


0.7478 


0.7490 


0.7501 


0.7513 


0.7524 


0.7536 


48 


cos 


0.6691 


6678 


0.6665 


0.6652 


0.6639 


0.6626 


0.6613 


0.6600 


0.6587 


0.6574 




tan 


1.1106 


1.1145 


1.1184 


I. 1224 


1 . 1263 


1.1303 


I. 1343 


1-1383 


I. 1423 


I . 1463 




sin 


0.7547 


0.7559 


0.7570 


0.7581 


0.7593 


0.7604 


0.761S 


0.7627 


0.7638 


0.7649 


49 


cos 


0.6561 


0.6547 


0.6534 


0.6521 


0.6508 


0.6494 


0.6481 


0.6468 


0.6455 


0.6441 




tan 


I . 1504 


I .1544 


I. 1585 


I . 1626 


I. 1667 


I . 1708 


I. 1750 


I. 1792 


I. 1833 


I. 1875 




sin 


0.7660 


0.7672 


0.7683 


0.7694 


0.770s 


0.7716 


0.7727 


0.7738 


0.7749 


0.7760 


50 


cos 


0.6428 


0.6414 


0.6401 


0.6388 


0.6374 


0.6361 


0.6347 


0.6334 


0.6320 


0.6307 




tan 


1.1918 


I . i960 


1.2002 


1.2045 


I . 2088 


1.2131 


I. 2174 


I. 2218 


I. 2261 


I. 230s 




sin 


0.7771 


0.7782 


0.7793 


0.7804 


0.7815 


0.7826 


0.7837 


0.7848 


0.7859 


0.7869 


51 


cos 


0.6293 


0.6280 


0.6266 


0.6252 


0.6239 


0.622s 


0.6211 


0.6198 


0.6184 


0.6170 




tan 


1.2349 


1.2393 


I . 2437 


I . 2482 


1.2527 


I . 2572 


I. 2617 


1.2662 


1.2708 


1.2753 




sin 


0.7880 


0.7891 


0.7902 


0.7912 


0.7923 


0.7934 


0.7944 


0.7955 


0.796s 


0.7976 


52 


cos 


0.6157 


0.6143 


0.6129 


0.6115 


0.6101 


0.6088 


0.6074 


0.6060 


0.6046 


0.6032 




tan 


1.2799 


1.2846 


1.2892 


1.2938 


1.2985 


1.3032 


1.3079 


I. 3127 


1.317s 


1.3222 




sin 


0.7986 


0.7997 


0.8007 


0.8018 


0.8028 


0.8039 


0.8049 


0.8059 


0.8070 


0.8080 


53 


cos 


0.6018 


0.6004 


0.5990 


0.5976 


0.5962 


0.5948 


O.S934 


0.5920 


0.5906 


0.5892 




tan 


1.3270 


I. 3319 


1.3367 


1.3416 


1.3465 


I. 3514 


1.3564 


I. 3613 


1.3663 


I -3713 




sin 


0.8090 


0.8100 


0.8111 


0.8121 


0.8131 


0.8141 


0.8151 


0.8161 


0.8171 


0.8181 


54 


cos 


0.S878 


0.5864 


0.5850 


0.5835 


0.5821 


0.5807 


0.5793 


0.5779 


0.5764 


0.5750 




tan 


1.3764 


I. 3814 


1.3865 


I. 3916 


1.3968 


I. 4019 


I. 4071 


I. 4124 


I. 4176 


1.4229 




sin 


0.8192 


0.8202 


0.8211 


0.8221 


0.8231 


0.8241 


0.8251 


0.8261 


0.8271 


0.8281 


55 


cos 


0.5736 


0.5721 


0.5707 


0.5693 


0.5678 


0.5664 


0.5650 


0.5635 


5621 


0.5606 




tan 


I. 4281 


1.4335 


1.4388 


I . 4442 


1.4496 


I . 4550 


1.4605 


1.4659 


1.4715 


1.4770 




sin 


0.8290 


0.8300 


0.8310 


0.8320 


0.8329 


0.8339 


0.8348 


0.8358 


0.8368 


0.8377 


56 


cos 


0.5592 


0.5577 


0.5563 


0.5548 


0.5534 


0.5519 


0.5505 


0.5490 


0.5476 


0.5461 




tan 


I . 4826 


I . 4882 


1.4938 


1.4994 


I. 5051 


I. 5108 


I. 5166 


1.5224 


1.5282 


1.5340 




sin 


0.8387 


08396 


0.8406 


0.8415 


0.8425 


0.8434 


0.8443 


0.8453 


0.8462 


0.8471 


57 


cos 


5446 


0.5432 


0.5417 


0.5402 


0.5388 


0.5373 


0.5358 


0.5344 


0.5329 


0.5314 




tan 


1.5399 


1.5458 


I 5517 


1.5577 


I . 5637 


1.5697 


1.5757 


I. 5818 


1.5880 


I. 5941 


58 


sin 


0.8480 


0.8490 


0.8499 


0.8508 


0.8517 


0.8526 


0.8536 


0.854s 


0.8554 


0.8563 


cos 


0.5299 


0.5284 


0.5270 


0.5255 


0.5240 


0.5225 


0.5210 


0.5195 


0.5180 


0.5165 




tan 


1.6003 


1.6066 


I. 6128 


1.6191 


1.6255 


I. 6319 


1.6383 


1.6447 


1.6512 


1.6577 


59 


sin 


0.8572 


8581 


0.8590 


0.8599 


0.8607 


0.8616 


0.8625 


0.8634 


0.8643 


0.8652 


cos 


0.5150 


0.5135 


0.5120 


0.5105 


0.5090 


0.5075 


0.5060 


0.5045 


0.5030 


0.5015 




tan 


1.6643 


1.6709 


1.6775 


1.6842 


1.6909 


1.6977 


1.7045 


I.7113 


I. 7182 


1.7251 


HDegs. 


Function 


0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 



24 



Natural Sines, Cosines and Tangents 























60 


'-74.9° 


Degs. 


Function 


o.o" 


0.1° 


0.2° 


0.3*" 


0.4° 


0.5° 


0.6" 


0.7° 


0.8° 


0.9° 




sin 


0.866c 


0.8669 


0.8678 


0.8686 


0.8695 


0.8704 


0.8712 


0.8721 


0.8729 


0.8738 


60 


cos 


o.sooc 


0.4985 


0.4970 


0.4955 


0.4939 


0.4924 


0.4909 


0.4894 


0.4879 


0.4863 




tan 


I. 7321 


I. 7391 


I. 7461 


1-7532 


1.7603 


1.7675 


1.7747 


1.7820 


1.7893 


1.7966 




sin 


0.8746 


0.8755 


0.8763 


0.8771 


0.8780 


0.8788 


0.8796 


0.8805 


0.8813 


0.8821 


61 


cos 


0.4848 


0.4833 


0.4818 


0.4802 


0.4787 


0.4772 


0.4756 


0.4741 


0.4726 


0.4710 




tan 


1.8040 


1-8115 


I. 8190 


1.8265 


I. 8341 


I. 8418 


1.8495 


1.8572 


1.8650 


1.8728 




sin 


0.8829 


0.8838 


0.8846 


0.8854 


0.8862 


0.8870 


0.8878 


0.8886 


0.8894 


0.8902 


62 


cos 


0.4695 


0.4679 


0.4664 


0.4648 


0.4633 


0.4617 


0.4602 


0.4586 


0.4571 


0.4555 




tan 


1.8807 


1.8887 


1.8967 


1.9047 


I. 9128 


I. 9210 


1.9292 


1-9375 


1.9458 


I -9542 




sin 


0.8910 


0.8918 


0.8926 


0.8934 


0.8942 


0.8949 


0.8957 


0.8965 


0.8973 


8980 


63 


cos 


0.4540 


0.4524 


0.4509 


0.4493 


0.4478 


0.4462 


0.4446 


0.4431 


0.4415 


0-4399 


1 


tan 


1.9626 


1.9711 


1.9797 


1.9883 


1.9970 


2.0057 


2.0145 


2.0233 


2.0323 


2.0413 


1 


sin 


0.8988 


0.8996 


0.9003 


0.9011 


0.9018 


0.9026 


0.9033 


0.9041 


0.9048 


0.9056 


M 


cos 


0.4384 


0.4368 


0.4352 


0.4337 


0.4321 


0.4305 


0.4289 


0.4274 


0.4258 


0.4242 




tan 


2.0503 


2.0594 


2.0686 


2.0778 


2.0872 


2.0965 


2.1060 


2.1155 


2.1251 


2.1348 




sin 


0.9063 


0.9070 


0.9078 


0.9085 


0.9092 


0.9100 


0.9107 


0.9114 


0.9121 


0.9128 


65 


cos 


0.4226 


0.4210 


0.4195 


0.4179 


0.4163 


0.4147 


0.4131 


0.4115 


0.4099 


0.4083 


i 


tan 


2.1445 


2.1543 


2.1642 


2.1742 


2.1842 


2.1943 


2.2045 


2.2148 


2.2251 


2.2355 


Ka 


sin 


0.9135 


0.9143 


0.9150 


0.9157 


0.9164 


0.9171 


0.9178 


0.9184 


0.9191 


0.9198 


66 


cos 


0.4067 


0.4051 


0.4035 


0.4019 


0.4003 


0.3987 


0.3971 


0.3955 


0.3939 


0.3923 




tan 


2.2460 


2.2566 


2.2673 


2.2781 


2 . 2889 


2.2998 


2.3109 


2.3220 


2.3332 


2.3445 




sin 


0.9205 


0.9212 


0.9219 


0.9225 


0.9232 


0.9239 


0.9245 


0.9252 


0.9259 


0.9265 


67 


cos 


0.3907 


0.3891 


0.3875 


0.3859 


0.3843 


0.3827 


0.3811 


0.3795 


0.3778 


0.3762 


"5 


tan 


2.3559 


2.3673 


2.3789 


2.3906 


2.4023 


2.4142 


2.4262 


2.4383 


2.4504 


2.4627 




sin 


0.9272 


0.9278 


0.9285 


0.9291 


0.9298 


0.9304 


0.9311 


0.9317 


0.9323 


0.9330 


68 


cos 


0.3746 


0.3730 


0.3714 


0.3697 


0.3681 


0.3665 


0.3649 


0.3633 


0.3616 


0.3600 




tan 


2.4751 


2.4876 


2.5002 


2.5129 


2.5257 


2.5386 


2.5517 


2.5649 


2.5782 


2.5916 




sin 


0.9336 


0.9342 


0.9348 


0.9354 


0.9361 


0.9367 


0.9373 


0.9379 


0.9385 


0.9391 


69 


cos 


0.3584 


0.3567 


0.3551 


0.3535 


0.3518 


0.3502 


0.3486 


0.3469 


0.3453 


0.3437 




tan 


2.6051 


2.6187 


2.6325 


2.6464 


2.6605 


2.6746 


2.6889 


2.7034 


2.7179 


2.7326 




sin 


0.9397 


0.9403 


0.9409 


0.9415 


0.9421 


0.9426 


0.9432 


0.9438 


0.9444 


0.9449 


70 


cos 


0.3420 


0.3404 


0.3387 


0.3371 


0.3355 


0.3338 


0.3322 


0.3305 


0.3289 


0.3272 




tan 


2.7475 


2.7625 


2.7776 


2.7929 


2.8083 


2.8239 


2.8397 


2.8556 


2.8716 


2.8878 




sin 


0.9455 


0.9461 


0.9466 


0.9472 


0.9478 


0.9483 


0.9489 


0,9494 


0.9500 


0.9505 


71 


cos 


0.3256 


0.3239 


0.3223 


0.3206 


0.3190 


0.3173 


0.3156 


0.3140 


0.3123 


0.3107 




tan 


2.9042 


2.9208 


2.9375 


2.9544 


2.9714 


2.9887 


3.0061 


3.0237 


3 0415 


3.0595 




sin 


0.9511 


0.9516 


0.9521 


0.9527 


0.9532 


0.9S37 


0.9542 


0.9548 


0.9553 


0.9558 


72 


cos 


0.3090 


0.3074 


0.3057 


0.3040 


0.3024 


0.3007 


0.2990 


0.2974 


0.2957 


0.2940 




tan 


3.0777 


3 0961 


3-1146 


3.1334 


3-1524 


3.1716 


3.1910 


3.2106 


3.2305 


3.2506 




sin 


0.9563 


0.9568 


0.9573 


0.9578 


0.9583 


0.9588 


0.9593 


0.9598 


0.9603 


0.9608 


73 


cos 


0.2924 


0.2907 


0.2890 


0.2874 


0.2857 


0.2840 


0.2823 


0.2807 


0.2790 


0.2773 




tan 


3.2709 


3.2914 


3-3122 


3.3332 


3-3544 


3.3759 


3.3977 


3.4197 


3.4420 


3.4646 




sin 


0.9613 


0.9617 


0.9622 


0.9627 


0.9632 


0.9636 


0.9641 


0.9646 


0.9650 


0.9655 


74 


cos . 


0.2756 


0.2740 


0.2723 


0.2706 


0.2689 


0.2672 


0.2656 


0.2639 


0.2622 


0.2605 




tan 


3-4874 


3-5105 


3-5339 


3.5576 


3-S816 


3.6059 


3.6305 


3.6554 


3.6806 


3-7062 


Degs. 


Function 


0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


S4' 



Natural Sines, Cosines and Tangents 



25 
75°-89.9° 



Degs. 


Function 


0.0° 


0.1° 


0.2° 


0.3° 


0.4° 


0.5° 


0.6° 


0.7° 


0.8° 


0.9° 




sin 


0.9659 


0.9664 


0.9668 


0.9673 


0.9677 


0.9681 


0.9686 


0.9690 


0.9694 


0.9699 


75 


cos 


0.2588 


0.2571 


0.2554 


0.2538 


0.2521 


0.2504 


0.2487 


0.2470 


0.2453 


0.2436 




tan 


3-7321 


3.7583 


3.7848 


3.8118 


3.8391 


3.8667 


3.8947 


3.9232 


3-9520 


3.9812 




sin 


0.9703 


0.9707 


9711 


0.9715 


0.9720 


0.9724 


0.9728 


0.9732 


0.9736 


0.9740 


76 


cos 


0.2419 


0.2402 


0.2385 


0.2368 


0.2351 


0.2334 


0.2317 


0.2300 


0.2284 


0.2267 




tan 


4.0108 


4.0408 


4.0713 


4.1022 


4.1335 


4.1653 


4.1976 


4-2303 


4-2635 


4.2972 




sin 


0.9744 


0.9748 


0.9751 


0.9755 


0.9759 


0.9763 


0.9767 


0.9770 


0.9774 


0.9778 


77 


cos 


0.2250 


0.2232 


0.2215 


0.2198 


0.2181 


0.2164 


0.2147 


0.2130 


0.2113 


0.2096 




tan 


4.3315 


4.3662 


4.4015 


4.4374 


4-4737 


4.5107 


4-5483 


4.5864 


4.6252 


4.6646 




sin 


0.9781 


0.9785 


0.9789 


0.9792 


0.9796 


0.9799 


0.9803 


0.9806 


0.9810 


0.9813 


78 


cos 


0.2079 


0.2062 


0.2045 


0.2028 


0.2011 


0.1994 


0.1977 


0.1959 


0.1942 


0.1925 




tan 


4.7046 


4-7453 


4.7867 


4.8288 


4.8716 


4.9152 


4-9594 


5.004s 


S.0504 


5. 0970 




sin 


0.9816 


0.9820 


0.9823 


0.9826 


0.9829 


0.9833 


0.9836 


0..9839 


0.9842 


0.9845 


79 


cos 


0.1908 


0.1891 


0.1874 


0.1857 


0.1840 


0.1822 


0.1805 


0.1788 


0.1771 


0.1754 




tan 


5. 1446 


5.1929 


5.2422 


5.2924 


5. 3435 


5.3955 


5.4486 


S.S026 


S.SS78 


5.6140 




sin 


0.9848 


0.9851 


0.9854 


0.9857 


0.9860 


0.9863 


0.9866 


0.9869 


0.9871 


0.9874 


80 


cos 


0.1736 


0.1719 


0.1702 


0.1685 


0.1668 


0.1650 


0.1633 


0.1616 


0.1599 


0.1582 




tan 


5.6713 


5.7297 


5.7894 


5.8502 


5.9124 


5.9758 


6.0405 


6.1066 


6.1742 


6.2432 


f 


sin 


0.9877 


0.9880 


0.9882 


0.9885 


0.9888 


0.9890 


0.9893 


0.9895 


0.9898 


0.9900 


81 


cos 


0.1564 


0.1547 


0.1530 


0.1513 


0.1495 


0.1478 


0.1461 


0.1444 


0.1426 


0.1409 




tan 


6.3138 


6.3859 


6.4596 


6.5350 


6.6122 


6.6912 


6.7720 


6.8548 


6.9395 


7.0264 




sin 


0.9903 


0.9905 


0.9907 


0.9910 


0.9912 


0.9914 


0.9917 


0.9919 


0.9921 


0.9923 


82 


cos 


0.1392 


0.1374 


0.1357 


0.1340 


0.1323 


0.1305 


0.1288 


0.1271 


O.I2S3 


0.1236 




tan 


7-1154 


7.2066 


7.3002 


7.3962 


7.4947 


7.5958 


7.6996 


7.8062 


7-9158 


8.028s 




sin 


0.9925 


0.9928 


0.9930 


0.9932 


0.9934 


0.9936 


0.9938 


0.9940 


0.9942 


0.9943 


83 


cos 


0.1219 


0.1201 


0.1184 


0.1167 


0.1149 


0.1132 


0.1115 


0.1097 


0.1080 


0.1063 




tan 


8.1443 


8.2636 


8.3863 


8.5126 


8.6427 


8.7769 


8.9152 


9-0579 


9.2052 


9-3572 




sin 


0.9945 


0.9947 


0.9949 


0.9951 


0.9952 


0.9954 


0.9956 


0.9957 


0.9959 


0.9960 


84 


cos 


0.1045 


0.1028 


O.IOII 


0.0993 


0.0976 


0.0958 


0.0941 


0.0924 


0.0906 


0.0889 




tan 


9.5144 


9.6768 


9.8448 


10.02 


10.20 


10.39 


10.58 


10.78 


10.99 


11.20 




sin 


0.9962 


0.9963 


0.9965 


0.9966 


0.9968 


0.9969 


0.9971 


0.9972 


0.9973 


0.9974 


85 


cos 


0.0872 


0.0854 


0.0837 


0.0819 


0.0802 


0.0785 


0.0767 


0.0750 


0.0732 


0.0715 




tan 


11.43 


11.66 


II. 91 


12.16 


12.43 


12.71 


13.00 


13-30 


13.62 


13.9s 




sin 


0.9976 


0.9977 


0.9978 


0.9979 


0.9980 


0.9981 


0.9982 


0.9983 


0.9984 


0.9985 


86 


cos 


0.0698 


0.0680 


0.0663 


0.0645 


0.0628 


0.0610 


0.0593 


0.0576 


0.0558 


0.0541 




tan 


14.30 


14.67 


15 06 


15.46 


IS. 89 


16.35 


16.83 


17.34 


17.89 


18.46 




sin 


0.9986 


0.9987 


0.9988 


0.9989 


0.9990 


0.9990 


0.9991 


0.9992 


0.9993 


0.9993 


87 


cos 


0.0523 


0.0506 


0.0488 


0.0471 


0.0454 


0.0436 


0.0419 


0.0401 


0.0384 


0.0366 




tan 


19.08 


19.74 


20.45 


21.20 


22.02 


22.90 


23.86 


24.90 


26.03 


27.27 




sin 


0.9994 


0.9995 


0.9995 


0.9996 


0.9996 


0.9997 


0.9997 


0.9997 


0.9998 


0.9998 


88 


cos 


0.0349 


0.0332 


0.0314 


0.0297 


0.0279 


0.0262 


0.0244 


0.0227 


0.0209 


0.0192 




tan 


28.64 


30.14 


31.82 


33.69 


35.80 


38.19 


40.92 


44-07 


47.74 


52.08 




sin 


0.9998 


0.9999 


0.9999 


0.9999 


0.9999 


1. 000 


1. 000 


1. 000 


1. 000 


1. 000 


89 


cos 


0.0175 


0.0157 


0.0140 


0.0122 


0.0105 


0.0087 


0.0070 


0.0052 


0.003s 


0.0017 




tan 


57.29 


63.66 


71.62 


81.85 


95.49 


114. 6 


143.2 


191. 


286.5 


573.0 


1 Degs. 


Function 


0' 


6' 


12' 


18' 


24' 


30' 


36' 


42' 


48' 


54' 



KB- 7.2 







^. 



.^''^-p 








^"'i-. 




0- -ms^^^' "^ov* 






/ -?► 




""-/ 'Mm,'. %^^^ ■ 







I*' 



^ « . o U ^ 

D08BS BROS. 

LIBRAKY BINOIjflGCi 










ST. AUGUSTINE ^ . • y-' 'b^ •■. rtT'" O'' ■^^ - . , , . 








c ^ 





•» J1 



